Two identities from one circle

Mathematics · Geometry · ages 16-17

Name ______________________ Date ____________

sin θ = 0.6 for acute θ. What is cos θ?
Answer: ______________
The unit circle has x^2 + y^2 = 1 with x = cos θ, y = sin θ. Which identity follows?
- sin^2 θ + cos^2 θ = 1
- sin θ + cos θ = 1
- sin^2 θ - cos^2 θ = 1
- 2 sin θ cos θ = 1
Which equation is the Pythagorean identity?
- Sin plus cos equals 1
- Tan equals sin times cos
- Sin squared plus cos squared equals 1
The Pythagorean identity holds only for acute angles.
Circle one: True False
What is sin^2 20° + cos^2 20°?
Answer: ______________
When verifying an identity, you may move terms across the equals sign.
Circle one: True False
Simplify (1 - cos^2 θ) / sin θ.
- cos θ
- tan θ
- 1
- sin θ
Maya says the identity comes from Pythagoras on the unit circle radius. Is Maya right?
Circle one: True False
A learner turns 1 minus cos squared into cos squared. What went wrong?
- Squares can be dropped freely
- The minus sign means divide
- The identity gives sin squared, not cos squared
sin θ = 0.28 for acute θ. What is cos θ?
Answer: ______________

Answer key

For grown-ups. Fold this page away before handing over the rest.

0.8 · cos squared = 1 − 0.36 = 0.64, root 0.8.
sin^2 θ + cos^2 θ = 1 · Direct substitution keeps the plus and the squares. The 2 sin θ cos θ option is the double-angle rule, not this.
Sin squared plus cos squared equals 1 · Squares come from x squared plus y squared on the unit circle.
False · Radius 1 proves it for every angle on the circle.
1 · The identity gives 1 for every angle.
False · Work on one side until it matches the other side.
sin θ · Top is sin squared; cancelling one sin leaves sin θ. The 1 option cancels too greedily.
True · Legs cos and sin with hypotenuse 1 give it.
The identity gives sin squared, not cos squared · Since the squares sum to 1, the remainder after cosine is sine.
0.96 · cos squared = 1 − 0.0784 = 0.9216, root 0.96.