Trigonometric identities and equations

On the unit circle, cosine is the horizontal coordinate. Usually two angles, reflected across the horizontal axis, have the same cosine.

1. Write the equation as cos x = c. If c is outside [−1, 1], there are no solutions.
2. Find α = arccos c between 0° and 180°. Use known angle values or a calculator in degree mode.
3. On [0°, 360°], check the angles α and 360° − α.
4. Keep all distinct solutions in the interval. If both angles coincide, list that angle only once; if you obtain 0° and 360°, both endpoints are included.

Check the sign: cosine is positive on the right half of the unit circle, negative on the left half, and zero at the points on the vertical axis.

Cosine equals one when its argument is an integer number of full turns. Solve for x, select values in the stated interval and add them. Add x values, not the inner-angle values.

Use tan α = sin α / cos α for cos α ≠ 0. Rearrange a sin x + b = 0 to sin x = −b/a for a ≠ 0. If this is outside [−1, 1], there is no solution. Otherwise use sine solutions and retain all in the specified interval.

Identify the appropriate sine or cosine sum or difference formula. Sine uses mixed products; cosine uses a cosine product and a sine product. Check the sign carefully.

Use tan α = sin α / cos α for cos α ≠ 0. Rearrange a sin x + b = 0 to sin x = −b/a for a ≠ 0. If this is outside [−1, 1], there is no solution. Otherwise use sine solutions and retain all in the specified interval.

Use the fundamental trigonometric identity:

sin² α + cos² α = 1.

1. Move sin² α to the other side: cos² α = 1 − sin² α.
2. Take the square root: cos α = √(1 − sin² α).
3. Because α is acute, its cosine is positive, so use the positive square root.
4. Round the result in the same way as the answer choices.

Use the double-angle sine formula. Multiply the given values, including the coefficient and signs, then reduce the fraction.

Choose the double-angle cosine formula involving only cos(α). Square the given value first, then complete the calculation and reduce the fraction.

Write all angles whose sine is zero. Rearrange kx + h° for x, dividing both the shift and the period term by the coefficient.

Recognise the double-angle sine identity or one of the cosine identities. Double the entire argument in parentheses, not just x.