Choose a representation
Reduce a cosine angle by full turns. Use sine odd symmetry for a negative angle. For a nonright triangle with two sides and included angle, use the cosine rule.
Worked example: follow the quantity
An angle θ satisfies sin θ=3/5. What is sin(−θ)? Sine is an odd function: reflecting the angle changes its sign. Keep the magnitude of the original sine value. Substitute the given numbers: -3 ÷ 5 = -3/5. Therefore the requested value is -3/5.
Contrast a plausible mistake
For that example, the tempting answer 9/25 follows this error: Square the original sine and ignore odd symmetry. The requested quantity is -3/5. Explain why the wrong operation changes the mathematical object, not merely the arithmetic.
A second representation
Two sides of a triangle are 9 and 24; their included angle is 60°. What is the third side? Use c²=9²+24²−2×9×24 cos 60°, with cos 60°=1/2. Take the positive square root after combining the three terms. c² = 9² + 24² − 2×9×24×(1/2) = 441; c = 21. Therefore the requested value is 21.
Transfer before checking
An angle θ satisfies sin θ=4/6. What is sin(−θ)? Try without viewing the answer. Check: -2/3. Sine is an odd function: reflecting the angle changes its sign. Keep the magnitude of the original sine value. Substitute the given numbers: -4 ÷ 6 = -2/3.
Scope and conditions
The sign of cosine depends on the quadrant even when its square is known. A full turn changes an angle but not its point on the unit circle. These original practice tasks sample the official topic; they do not certify complete coverage or a predicted ACT score.
An angle θ satisfies sin θ=3/5. What is sin(−θ)?
- Sine is an odd function: reflecting the angle changes its sign.
- Keep the magnitude of the original sine value.
- Substitute the given numbers: -3 ÷ 5 = -3/5.
- Therefore the requested value is -3/5.
Compare the other answer choices
Square the original sine and ignore odd symmetry.
Treat sine as an even function.
Square the sine before negating it.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
ACT Mathematics content