Question

Fill in the blank with “all”, “no”, or “some” to make the following statements true. Note that “some” means one or more instances, but not all.
•If your answer is “all”, then give a brief explanation as to why.
•If your answer is “no”, then give an example and a brief explanation as to why.
•If your answer is “some”, then give two specific examples that illustrate why your answer it not “all” or “no”. Be sure to explain your two examples.
An example must include either a graph or a specific function.

1. Let θ and φ be angles such that θ +φ = π/2 radians. For ______ such angles, it follows that cos(θ)= sin(φ).

Answer 1

They are talking about complimentary angles, that is,
θ +φ = 90°
then for all cases,
cos(θ)= sin(φ)

e.g. cos 70° = sin 20°
cos 12.34° = 77.66°
etc

sketch a right-angled triangle with angles θ and φ
now take cos(θ) and sin(φ).
You will get the same ratio in terms of opposite, adjacent and hypotenuse