Question 1

Which negative angle identities does the explorer cover?

Accepted Answer

All six standard identities: sin(-theta) = -sin theta, cos(-theta) = cos theta, tan(-theta) = -tan theta, csc(-theta) = -csc theta, sec(-theta) = sec theta, and cot(-theta) = -cot theta. Each has its own tab with either a geometric proof or an algebraic derivation.

Question 2

Which trig functions are even and which are odd?

Accepted Answer

Cosine and secant are even, meaning f(-theta) equals f(theta). Sine, tangent, cosecant, and cotangent are odd, meaning f(-theta) equals minus f(theta). The Parity column of the formula table labels each one.

Question 3

How is sin(-theta) = -sin theta proved geometrically?

Accepted Answer

Place the terminal point P of angle theta on the unit circle, then reflect across the x-axis to obtain P prime, which is the terminal point of -theta. The reflection flips the y-coordinate but preserves the x-coordinate, so sin(-theta) equals -sin theta and cos(-theta) equals cos theta.

Question 4

Why is secant even but cosecant odd?

Accepted Answer

A reciprocal preserves parity. Cosine is even, so secant equal to 1 over cosine is also even. Sine is odd, so cosecant equal to 1 over sine is odd. The same logic makes tangent odd and cotangent odd.

Question 5

Can I verify an identity at a specific angle?

Accepted Answer

Yes. Move the theta slider to the angle you want, then read the two metric cards near the bottom. They show the function value at plus theta and at minus theta. For odd functions the signs differ, for even functions they match. The formula table does the same for all six functions at once.

Negative Angle Trigonometric Identities