Use a sum or difference identity to find the EXACT value: sin ⁡ ( 15 ° ) \sin\left(15\degree\right) sin ( 1 5 ° )

6 − 2 4 \frac{\sqrt{6}-\sqrt{2}}{4} 4 6 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> − 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

2 − 6 4 \frac{\sqrt{2}-\sqrt{6}}{4} 4 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> − 6 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

Use a sum or difference identity to find the EXACT value: sin ⁡ ( 75 ° ) \sin\left(75\degree\right) sin ( 7 5 ° )

2 + 6 4 \frac{\sqrt{2}+\sqrt{6}}{4} 4 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> + 6 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

2 − 6 4 \frac{\sqrt{2}-\sqrt{6}}{4} 4 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> − 6 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

Use a sum or difference identity to find the EXACT value: cos ⁡ ( 75 ° ) \cos\left(75\degree\right) cos ( 7 5 ° )

6 − 2 4 \frac{\sqrt{6}-\sqrt{2}}{4} 4 6 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> − 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

2 − 6 4 \frac{\sqrt{2}-\sqrt{6}}{4} 4 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> − 6 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

Use a sum or difference identity to find the EXACT value: cos ⁡ ( 105 ° ) \cos\left(105\degree\right) cos ( 1 0 5 ° )

2 − 6 4 \frac{\sqrt{2}-\sqrt{6}}{4} 4 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> − 6 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

6 − 2 4 \frac{\sqrt{6}-\sqrt{2}}{4} 4 6 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> − 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

Use a sum or difference identity to find the EXACT value: cos ⁡ ( π 12 ) \cos\left(\frac{\pi}{12}\right) cos ( 1 2 π )

2 + 6 4 \frac{\sqrt{2}+\sqrt{6}}{4} 4 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> + 6 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

2 − 6 4 \frac{\sqrt{2}-\sqrt{6}}{4} 4 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> − 6 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

Use a sum or difference identity to find the EXACT value: sin ⁡ ( 7 π 12 ) \sin\left(\frac{7\pi}{12}\right) sin ( 1 2 7 π )

6 + 2 4 \frac{\sqrt{6}+\sqrt{2}}{4} 4 6 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> + 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

6 − 2 4 \frac{\sqrt{6}-\sqrt{2}}{4} 4 6 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> − 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

Use a sum or difference identity to find the EXACT value: cos ⁡ ( 7 π 12 ) \cos\left(\frac{7\pi}{12}\right) cos ( 1 2 7 π )

2 − 6 4 \frac{\sqrt{2}-\sqrt{6}}{4} 4 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> − 6 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

6 − 2 4 \frac{\sqrt{6}-\sqrt{2}}{4} 4 6 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z"> − 2 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5, -10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8, -50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0, 35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5, -221c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467 s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422 s-65,47,-65,47z M834 80H400000v40H845z">

Write the expression as the sine or cosine of a single angle: sin ⁡ 42 ° cos ⁡ 17 ° − cos ⁡ 42 ° sin ⁡ 17 ° \sin42\degree\cos17\degree-\cos42\degree\sin17\degree sin 4 2 ° cos 1 7 ° − cos 4 2 ° sin 1 7 °

sin ⁡ ( 25 ° ) \sin\left(25\degree\right) sin ( 2 5 ° )

sin ⁡ ( 59 ° ) \sin\left(59\degree\right) sin ( 5 9 ° )

cos ⁡ ( 59 ° ) \cos\left(59\degree\right) cos ( 5 9 ° )

cos ⁡ ( 25 ° ) \cos\left(25\degree\right) cos ( 2 5 ° )

Write the expression as the sine or cosine of a single angle: cos ⁡ 94 ° cos ⁡ 18 ° + sin ⁡ 94 ° sin ⁡ 18 ° \cos94\degree\cos18\degree+\sin94\degree\sin18\degree cos 9 4 ° cos 1 8 ° + sin 9 4 ° sin 1 8 °

cos ⁡ ( 76 ° ) \cos\left(76\degree\right) cos ( 7 6 ° )

sin ⁡ ( 76 ° ) \sin\left(76\degree\right) sin ( 7 6 ° )

cos ⁡ ( 112 ° ) \cos\left(112\degree\right) cos ( 1 1 2 ° )

sin ⁡ ( 112 ° ) \sin\left(112\degree\right) sin ( 1 1 2 ° )

Write the expression as the sine or cosine of a single angle: sin ⁡ ( π 5 ) cos ⁡ ( π 2 ) + sin ⁡ ( π 2 ) cos ⁡ ( π 5 ) \sin\left(\frac{\pi}{5}\right)\cos\left(\frac{\pi}{2}\right)+\sin\left(\frac{\pi}{2}\right)\cos\left(\frac{\pi}{5}\right) sin ( 5 π ) cos ( 2 π ) + sin ( 2 π ) cos ( 5 π )

sin ⁡ ( 7 π 10 ) \sin\left(\frac{7\pi}{10}\right) sin ( 1 0 7 π )

cos ⁡ ( 3 π 10 ) \cos\left(\frac{3\pi}{10}\right) cos ( 1 0 3 π )

sin ⁡ ( 3 π 10 ) \sin\left(\frac{3\pi}{10}\right) sin ( 1 0 3 π )

cos ⁡ ( 7 π 10 ) \cos\left(\frac{7\pi}{10}\right) cos ( 1 0 7 π )

Write the expression as the sine or cosine of a single angle: sin ⁡ ( π 3 ) cos ⁡ ( π 7 ) − sin ⁡ ( π 7 ) cos ⁡ ( π 3 ) \sin\left(\frac{\pi}{3}\right)\cos\left(\frac{\pi}{7}\right)-\sin\left(\frac{\pi}{7}\right)\cos\left(\frac{\pi}{3}\right) sin ( 3 π ) cos ( 7 π ) − sin ( 7 π ) cos ( 3 π )

sin ⁡ ( 4 π 21 ) \sin\left(\frac{4\pi}{21}\right) sin ( 2 1 4 π )

cos ⁡ ( 4 π 21 ) \cos\left(\frac{4\pi}{21}\right) cos ( 2 1 4 π )

cos ⁡ ( 10 π 21 ) \cos\left(\frac{10\pi}{21}\right) cos ( 2 1 1 0 π )

sin ⁡ ( 10 π 21 ) \sin\left(\frac{10\pi}{21}\right) sin ( 2 1 1 0 π )

Write the expression as the sine or cosine of a single angle: cos ⁡ ( π 7 ) cos ⁡ x + sin ⁡ ( π 7 ) sin ⁡ x \cos\left(\frac{\pi}{7}\right)\cos x+\sin\left(\frac{\pi}{7}\right)\sin x cos ( 7 π ) cos x + sin ( 7 π ) sin x

cos ⁡ ( π 7 − x ) \cos\left(\frac{\pi}{7}-x\right) cos ( 7 π − x )

sin ⁡ ( π 7 − x ) \sin\left(\frac{\pi}{7}-x\right) sin ( 7 π − x )

cos ⁡ ( π 7 + x ) \cos\left(\frac{\pi}{7}+x\right) cos ( 7 π + x )

sin ⁡ ( π 7 + x ) \sin\left(\frac{\pi}{7}+x\right) sin ( 7 π + x )

Write the expression as the sine or cosine of a single angle: cos ⁡ x cos ⁡ 15 ° − sin ⁡ x sin ⁡ 15 ° \cos x\cos15\degree-\sin x\sin15\degree cos x cos 1 5 ° − sin x sin 1 5 °

cos ⁡ ( x + 15 ° ) \cos\left(x+15\degree\right) cos ( x + 1 5 ° )

cos ⁡ ( x − 15 ° ) \cos\left(x-15\degree\right) cos ( x − 1 5 ° )

sin ⁡ ( x − 15 ° ) \sin\left(x-15\degree\right) sin ( x − 1 5 ° )

sin ⁡ ( x + 15 ° ) \sin\left(x+15\degree\right) sin ( x + 1 5 ° )

Write the expression as the sine or cosine of a single angle: sin ⁡ ( 3 x ) cos ⁡ x − cos ⁡ ( 3 x ) sin ⁡ x \sin\left(3x\right)\cos x-\cos\left(3x\right)\sin x sin ( 3 x ) cos x − cos ( 3 x ) sin x

sin ⁡ ( 2 x ) \sin\left(2x\right) sin ( 2 x )

cos ⁡ ( 2 x ) \cos\left(2x\right) cos ( 2 x )

cos ⁡ ( 4 x ) \cos\left(4x\right) cos ( 4 x )

sin ⁡ ( 4 x ) \sin\left(4x\right) sin ( 4 x )

Write the expression as the sine or cosine of a single angle: cos ⁡ ( 7 y ) cos ⁡ ( 3 y ) − sin ⁡ ( 7 y ) sin ⁡ ( 3 y ) \cos\left(7y\right)\cos\left(3y\right)-\sin\left(7y\right)\sin\left(3y\right) cos ( 7 y ) cos ( 3 y ) − sin ( 7 y ) sin ( 3 y )

cos ⁡ ( 10 y ) \cos\left(10y\right) cos ( 1 0 y )

cos ⁡ ( 4 y ) \cos\left(4y\right) cos ( 4 y )

sin ⁡ ( 4 y ) \sin\left(4y\right) sin ( 4 y )

sin ⁡ ( 10 y ) \sin\left(10y\right) sin ( 1 0 y )

Angle Sum/Difference Identities

Authored by Adam Hosler

Mathematics

10th - 12th Grade

CCSS covered

Used 119+ times

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20 questions

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MULTIPLE CHOICE QUESTION

3 mins • 1 pt

Only using angles available on the unit circle, find the set of angles that will add or subtract to the given angle:
$15\degree$

$75\degree-60\degree$

$115\degree-100\degree$

$60\degree-45\degree$

$85\degree-70\degree$

Tags

CCSS.HSG.SRT.C.8

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

Only using angles available on the unit circle, find the set of angles that will add or subtract to the given angle:
$75\degree$

$80\degree-5\degree$

$40\degree+35\degree$

$100\degree-25\degree$

$30\degree+45\degree$

Tags

CCSS.HSG.SRT.C.8

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

Only using angles available on the unit circle, find the set of angles that will add or subtract to the given angle:
$165\degree$

$185\degree-20\degree$

$45\degree+120\degree$

$35\degree+130\degree$

$180\degree-15\degree$

Tags

CCSS.HSF.TF.A.3

CCSS.HSF.TF.A.4

CCSS.HSF.TF.C.8

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

Only using angles available on the unit circle, find the set of angles that will add or subtract to the given angle:
$105\degree$

$120\degree-15\degree$

$180\degree-75\degree$

$55\degree+50\degree$

$60\degree+45\degree$

Tags

CCSS.HSF.TF.A.3

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

Only using angles available on the unit circle, find the set of angles that will add or subtract to the given angle:
$255\degree$

$200\degree+55\degree$

$220\degree+35\degree$

$300\degree-45\degree$

$360\degree-105\degree$

Tags

CCSS.HSF.TF.C.9

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

Use a sum or difference identity to find the EXACT value:

$\sin\left(15\degree\right)$

0.2588

0.6503

$\frac{\sqrt{6}-\sqrt{2}}{4}$

$\frac{\sqrt{2}-\sqrt{6}}{4}$

Tags

CCSS.HSF.TF.C.9

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

Use a sum or difference identity to find the EXACT value:
$\sin\left(75\degree\right)$

-0.3879

0.9659

$\frac{\sqrt{2}-\sqrt{6}}{4}$

$\frac{\sqrt{2}+\sqrt{6}}{4}$

Tags

CCSS.HSF.TF.C.9

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

Use a sum or difference identity to find the EXACT value:
$\cos\left(75\degree\right)$

0.2588

0.9218

$\frac{\sqrt{6}-\sqrt{2}}{4}$

$\frac{\sqrt{2}-\sqrt{6}}{4}$

Tags

CCSS.HSF.TF.C.9

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

Use a sum or difference identity to find the EXACT value:

$\cos\left(105\degree\right)$

-0.2410

-0.2588

$\frac{\sqrt{6}-\sqrt{2}}{4}$

$\frac{\sqrt{2}-\sqrt{6}}{4}$

Tags

CCSS.HSF.TF.C.9

10.

MULTIPLE CHOICE QUESTION

3 mins • 1 pt

Use a sum or difference identity to find the EXACT value:

$\cos\left(\frac{\pi}{12}\right)$

0.9659

0.9999

$\frac{\sqrt{2}+\sqrt{6}}{4}$

$\frac{\sqrt{2}-\sqrt{6}}{4}$

Tags

CCSS.HSF.TF.C.9

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Angle Sum/Difference Identities

Only using angles available on the unit circle, find the set of angles that will add or subtract to the given angle: 15°15\degree15°

Only using angles available on the unit circle, find the set of angles that will add or subtract to the given angle: 75°75\degree75°

Only using angles available on the unit circle, find the set of angles that will add or subtract to the given angle: 165°165\degree165°

Only using angles available on the unit circle, find the set of angles that will add or subtract to the given angle: 105°105\degree105°

Only using angles available on the unit circle, find the set of angles that will add or subtract to the given angle: 255°255\degree255°

Use a sum or difference identity to find the EXACT value: sin⁡(15°)\sin\left(15\degree\right)sin(15°)

Use a sum or difference identity to find the EXACT value: sin⁡(75°)\sin\left(75\degree\right)sin(75°)

Use a sum or difference identity to find the EXACT value: cos⁡(75°)\cos\left(75\degree\right)cos(75°)

Use a sum or difference identity to find the EXACT value: cos⁡(105°)\cos\left(105\degree\right)cos(105°)

Use a sum or difference identity to find the EXACT value: cos⁡(π12)\cos\left(\frac{\pi}{12}\right)cos(12π​)

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Only using angles available on the unit circle, find the set of angles that will add or subtract to the given angle:
$15\degree$

Only using angles available on the unit circle, find the set of angles that will add or subtract to the given angle:
$75\degree$

Only using angles available on the unit circle, find the set of angles that will add or subtract to the given angle:
$165\degree$

Only using angles available on the unit circle, find the set of angles that will add or subtract to the given angle:
$105\degree$

Only using angles available on the unit circle, find the set of angles that will add or subtract to the given angle:
$255\degree$

Use a sum or difference identity to find the EXACT value:

$\sin\left(15\degree\right)$

Use a sum or difference identity to find the EXACT value:
$\sin\left(75\degree\right)$

Use a sum or difference identity to find the EXACT value:
$\cos\left(75\degree\right)$

Use a sum or difference identity to find the EXACT value:

$\cos\left(105\degree\right)$

Use a sum or difference identity to find the EXACT value:

$\cos\left(\frac{\pi}{12}\right)$