In Example 6, change 180° + x to 180° − x and then determine what other changes result.

Data from Example 6

Prove that sin(180° + x) = −sin x.

Although x may or may not be an acute angle, this agrees with the results for the sine of a third-quadrant angle as discussed in Section 8.2. See Fig. 20.8. The calculator check of this identity, with y₁ = sin(π + x) and y₂ = −sin x is shown in Fig. 20.9. This screen was captured while y₂ (in red) was being graphed right over y₁ (in blue).

sin(180 + x) = sin180cosx + cos180sinx using Eq. (20.9) sin 180 0, cos 180 = -1 = (0)cosx +(1)sinx =sinx =