Trigonometric Identities and Application

$$=\sec\theta\cos\theta$$   $$\sec\theta\sqrt[]{1-\sin^2\theta}$$  

$$=\sec\theta\sqrt[]{\cos^2\theta}$$   $$=\sec\theta\cos\theta$$   $$=\frac{1}{\cos\theta}\cdot\cos\theta$$

$$=1$$  

$$\left(1-\cos^2\theta\right)\csc^2\theta$$   $$=\sin^2\theta\csc^2\theta$$   $$=\sin^2\theta\cdot\frac{1}{\sin^2\theta}$$

$$=1$$  

$$\tan^4\theta+\tan^2\theta$$ $$=\tan^2\theta\left(\tan^2\theta+1\right)$$

$$=\left(\sec^2\theta-1\right)\left(\sec^2\theta\right)$$ $$=\sec^4\theta-\sec^2\theta$$  

1. $$90°-28°\ =\ 62°$$

$$\cos62°=\frac{1200}{d}$$ , where d is the distance of the plane to the rock

$$d=2556.07m$$  

2. $$\sin28°=\frac{1200m}{d}$$  

$$d=\frac{1200m}{\sin28°}$$   $$d=2556.07m$$  

$$\sin52°=\frac{x}{100ft}$$ , x is the height of the kite to the ground

$$100ft\cdot\sin52°=x$$

$$x=100\cdot\sin52°$$

$$x\approx78.80ft$$  

$$\tan\theta=\frac{60}{200}$$   $$\theta=\tan^{-1}\left(\frac{60}{200}\right)$$  

$$\theta\approx16.699°\approx16.70°$$  

$$\sin\theta=\frac{2}{5}$$

$$\theta=\sin^{-1}\left(\frac{2}{5}\right)$$  

$$\theta\approx23.58°\ <30°$$