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Cogito

Trigonometry · Chapter 5 · Lesson 2

Practice Worksheet

Proving an Identity

Work one side until it turns into the other.

12 problems · about 22 minutes · F-TF.C.8

NameDate

Figure — use these to answer the problems

Move the angle and watch the two coordinates trade size while their squares still total 1.

Angle55° = π/3 rad
x-coordinate0.574
y-coordinate0.819
cos 55°0.574
sin 55°0.819

Angle in degrees55

Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

What is sin²θ + cos²θ for any angle?
Answer
2
Why may you not move terms across an identity you are proving?
- It assumes the equation is true, which is the thing being proved.
- It makes the algebra harder.
sin²θ = 0.49. What is cos²θ?
Answer
tan²θ + 1 = 9. What is sec²θ?
Answer
cot²θ = 15. What is csc²θ?
Answer

Build It Up

The same ideas with more to keep track of.

3 problems

6
Match each expression with its simplest form.
Draw a line from each item on the left to its match on the right.
- sin θ · csc θ
- sin θ ÷ cos θ
- 1 − cos²θ
- 1
- tan θ
- sin²θ
cos θ · sec θ equals what number?
Answer
sec²θ − tan²θ equals what number?
Answer

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

4 problems

9
Put the proof of tan θ · cos θ = sin θ in order.
Write 1 to 4 in the boxes to put these in order.
- Start with the left side, tan θ · cos θ.
- Replace tan θ with sin θ ÷ cos θ.
- Cancel the cos θ terms.
- The left side is now sin θ, matching the right.
cos θ = 0.8 and θ is acute. What is sin θ?
Answer
The Substitution
If sin²θ = 0.36, what is cos²θ?
Answer
12
The First Move
To prove sec θ · cot θ = csc θ, what is the best opening move?
- Rewrite the left side as sine and cosine.
- Multiply both sides by sin θ.
- Square both sides.