Negative Angles and Their Trigonometric Ratios
Simple Explanation
A negative angle is simply measured clockwise instead of the usual counterclockwise direction. Reflecting an angle θ to −θ flips the sign of y (its sine) but leaves x (its cosine) unchanged, giving: sin(−θ) = −sin θ, cos(−θ) = cos θ, and tan(−θ) = −tan θ.
Why Do We Need It?
Negative angles come up constantly in navigation, rotation, and later calculus — being able to relate them straight back to the equivalent positive angle avoids re-deriving everything from scratch.
See It
Two rays from the origin, one at 60 degrees above the x-axis and one at 60 degrees below it, mirror images of each other
Formula
Negative Angle Identities
sin(−θ) = −sin θ, cos(−θ) = cos θ, tan(−θ) = −tan θ
A negative angle is measured clockwise instead of counterclockwise. Sine and tangent of a negative angle are the negative of the positive angle's value; cosine is unchanged.
- −θ
- — the angle θ measured in the clockwise direction instead
When to use it: Whenever an angle is given as negative (measured clockwise), to rewrite it in terms of the equivalent positive angle's ratios.
Worked Example
Evaluate a negative angle's sine
Given sin 40° ≈ 0.643, find sin(−40°).
Why Does This Work?
Reflecting a point (x, y) across the x-axis (which is exactly what measuring the angle clockwise instead of counterclockwise does) sends y to −y while leaving x unchanged. Since sin θ=y/r and cos θ=x/r, this directly explains why sine flips sign while cosine does not; tan θ=y/x then also flips sign, since only its numerator changes sign.
Real-Life Example
A ship turning left versus right from its heading
A ship turns 40° clockwise ("right") from its current heading, versus 40° counterclockwise ("left").
These two turns correspond to angles θ and −θ, and the negative angle identities let a navigator relate the resulting readings directly, without separately recalculating each one.
Practice
If cos 25° ≈ 0.906, find cos(−25°).
MediumCommon mistake
Assuming cos(−θ) also flips sign, like sine and tangent do — only sine and tangent flip sign for a negative angle; cosine stays exactly the same.
Quick Review
- sin(−θ) = −sin θ, cos(−θ) = cos θ, tan(−θ) = −tan θ.
- A negative angle is measured clockwise instead of counterclockwise.
- Reflecting across the x-axis flips y (and hence sin, tan) but not x (cos).