HPC Section 5.1 Part 1

Given the diagram, which of the following statements are true?

 Which of the following would be the definition of

$$\cos\theta$$  ?

 Which of the following would be the definition of
$$\cot\theta$$  ?

 Which of the following would be the definition of
$$\sec\theta$$  ?

 Which of the following would be the definition of
$$\csc\theta$$  ?

Use

$$\csc x=2$$

find the other trig functions.  sin x = ?

Use

$$\csc x=2$$  

find the other trig functions.  If x is in the first quadrant what is cos x = ?

Use

$$\csc x=2$$  

find the other trig functions.  If x is in the first quadrant what is sec x = ?

Use

$$\csc x=2$$

 find the other trig functions.  If x is in the first quadrant what is tan x = ?

Rearrange:

  $$\sin^2x+\cos^2x=1$$  to solve for:   $$\sin^2x$$   

Rearrange: 

$$\sin^2x+\cos^2x=1$$  to solve for  $$\cos^2x$$  

Rearrange to solve for $$\tan^2\theta$$ : 
                                        $$1\ +\ \tan^2\theta=\ \sec^2\theta$$  

Rearrange so it equals 1: 
                                        $$1\ +\ \tan^2x=\ \sec^2x$$  

Solve the following for  $$\cot^2\theta$$  :
$$1+\cot^2\theta=\csc^2\theta$$  

$$\sin^2\theta+\cos^2\theta=1$$  

Choose the equivalent equations?

$$1+\ \tan^2\theta=\sec^2\theta$$  

Choose the equivalent equations?

Exchange each term with an identity then use algebra to reduce:

  $$\frac{\tan x}{\sec x}$$

Now try this one.  Again reduce by substituting with identities:
$$\left(\sec x\right)\left(\cot x\right)\left(\sin x\right)$$

Reduce by substituting with identities:

$$\tan\ x\ \cot x\ -\cos^2x$$  

Simplify by factoring out a GCF first. Look at what you have left.  Can you replace with an identity?

$$\cos^3x+\sin^2x\cos x$$

Distribute then exchange with identities:
$$\sin x\left(\sec x-\csc x\right)$$

Substitute then find a common denominator so you can combine into 1 fraction:              $$\cos x+\sin x\tan x$$  

Simplify by first splitting into 2 fractions: (Note:  you CANNOT cross out anything until you've done this!)

$$\frac{\cos x-\sin x}{\sin x\cos x}$$