why does sin 10 degrees and cos 80 degrees equal

why does sin 10 degrees and cos 80 degrees equal

The correct answer and explanation is:

Correct Answer:
sin⁡10∘=cos⁡80∘\sin 10^\circ = \cos 80^\circ
Both are approximately equal to 0.1736.

Explanation:
Sine and cosine are trigonometric functions that describe the relationships between angles and side lengths in a right triangle. A key identity in trigonometry is: sin⁡θ=cos⁡(90∘−θ)\sin \theta = \cos (90^\circ – \theta)

This is known as a co-function identity. It tells us that the sine of an angle is equal to the cosine of its complement. Two angles are complementary if their measures add up to 90 degrees. In this case: 10∘+80∘=90∘10^\circ + 80^\circ = 90^\circ

Therefore, since 80 degrees is the complement of 10 degrees: sin⁡10∘=cos⁡(90∘−10∘)=cos⁡80∘\sin 10^\circ = \cos (90^\circ – 10^\circ) = \cos 80^\circ

This identity is derived from the geometric definition of sine and cosine in the unit circle. In a unit circle, the sine of an angle is the y-coordinate of the point on the circle at that angle, and the cosine is the x-coordinate. When the angle increases or decreases by a complementary amount, the roles of x and y are switched, which creates the identity.

In practical calculations using a calculator or trigonometric table: sin⁡10∘≈0.1736\sin 10^\circ \approx 0.1736 cos⁡80∘≈0.1736\cos 80^\circ \approx 0.1736

This shows the values are indeed equal, confirming the co-function identity. Understanding this identity helps solve problems involving right triangles and simplifies expressions in trigonometry. It also demonstrates how angles in different parts of a triangle relate to one another. This identity is part of a larger system of relationships that include tangent, secant, cosecant, and cotangent.