Step by Step
1
Cosine is even
cos(−θ) = cos(θ) — plugging in a negative angle gives the exact same value as the positive angle. This is because cosine's graph is symmetric about the y-axis.
2
Sine is odd
sin(−θ) = −sin(θ) — plugging in a negative angle flips the sign of the result. This is because sine's graph is symmetric about the origin (rotationally symmetric).
3
Tangent is odd
tan(−θ) = −tan(θ), following the same odd-function pattern as sine (which makes sense, since tangent is defined as sine divided by cosine, and an odd function divided by an even function produces an odd function).
4
The geometric meaning of even vs. odd
"Even" means the function's graph is symmetric about the y-axis (mirror image on both sides). "Odd" means the graph has 180° rotational symmetry about the origin.
Applied Walkthrough
1
Simplify sin(−30°) using the even-odd identities. Since sine is an odd function, sin(−30°) = −sin(30°).
2
Recall from special right triangles that sin(30°) = 1/2.
3
So sin(−30°) = −(1/2) = −1/2.
4
By contrast, cos(−30°) = cos(30°) = √3/2 (unchanged), since cosine is even — illustrating the key difference between how sine and cosine behave under negation.
Exam Application
Exams test whether you know which trig functions are even (cosine) versus odd (sine, tangent), and whether you can correctly simplify expressions involving negative angles using these identities.
⚠ Common Trap
The most common trap is applying the odd-function sign flip to cosine, or forgetting to apply it to sine — remember cosine is the one exception that stays unchanged under negation; sine and tangent both flip sign.
✓ Quick Self-Check
1. Is cosine an even or odd function?
Even — cos(−θ) = cos(θ).
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2. Is sine an even or odd function?
Odd — sin(−θ) = −sin(θ).
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3. Is tangent an even or odd function?
Odd — tan(−θ) = −tan(θ).
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4. Simplify sin(−30°).
−1/2.
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5. What does it mean geometrically for a function to be "even"?
Its graph is symmetric about the y-axis.
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