Given: cos 20° = p. Without using p, find:
1. tan 70°?
2. sin 40°?
3. sin 50°?

To tackle this problem, we start by using the given information that cos 2 0 ∘ = p . We need to find expressions for tan 7 0 ∘ , sin 4 0 ∘ , and sin 5 0 ∘ without directly using the variable p .

Finding tan 7 0 ∘ :
We know that tan ( 9 0 ∘ − θ ) = cot θ . Therefore,
tan 7 0 ∘ = cot ( 9 0 ∘ − 7 0 ∘ ) = cot 2 0 ∘ .
cot 2 0 ∘ is the reciprocal of tan 2 0 ∘ . Therefore, we can write:
tan 7 0 ∘ = tan 2 0 ∘ 1 ​ .

Finding sin 4 0 ∘ :
Using the sine complement identity sin ( 9 0 ∘ − θ ) = cos θ , we have:
sin 4 0 ∘ = cos ( 9 0 ∘ − 4 0 ∘ ) = cos 5 0 ∘ .

Finding sin 5 0 ∘ :
Again, using the sine complement identity,
sin 5 0 ∘ = cos ( 9 0 ∘ − 5 0 ∘ ) = cos 4 0 ∘ .


In summary, you relied on trigonometric identities to find the relationships:

tan 7 0 ∘ = t a n 2 0 ∘ 1 ​

sin 4 0 ∘ = cos 5 0 ∘

sin 5 0 ∘ = cos 4 0 ∘


This approach helps you find the necessary trigonometric functions without directly using p = cos 2 0 ∘ .

Answered by ElijahBenjaminCarter | 2025-07-06