19. if $\\sin\\theta =.6000$, find the value of $\\sin(180^{\\circ}+\\theta)$. 21. if $\\cos25^{\\circ}=.9063…

Explanation:

Step1: Use trigonometric identity

For ( \sin(180^{\circ}+\theta)), use the identity ( \sin(A + B)=\sin A\cos B+\cos A\sin B). Here (A = 180^{\circ}), (B=\theta). So ( \sin(180^{\circ}+\theta)=\sin180^{\circ}\cos\theta+\cos180^{\circ}\sin\theta). Since ( \sin180^{\circ}=0) and ( \cos180^{\circ}=- 1), then ( \sin(180^{\circ}+\theta)=-\sin\theta). Given ( \sin\theta = 0.6000), so ( \sin(180^{\circ}+\theta)=-0.6000).

Step2: For ( \cos155^{\circ})

Use the identity ( \cos(A + B)=\cos A\cos B-\sin A\sin B). We can write (155^{\circ}=180^{\circ}-25^{\circ}). So ( \cos155^{\circ}=\cos(180^{\circ}-25^{\circ})). Using the identity ( \cos(A - B)=\cos A\cos B+\sin A\sin B) with (A = 180^{\circ}), (B = 25^{\circ}), we get ( \cos(180^{\circ}-25^{\circ})=\cos180^{\circ}\cos25^{\circ}+\sin180^{\circ}\sin25^{\circ}). Since ( \sin180^{\circ}=0) and ( \cos180^{\circ}=-1), then ( \cos155^{\circ}=-\cos25^{\circ}). Given ( \cos25^{\circ}=0.9063), so ( \cos155^{\circ}=- 0.9063).

Answer:

For ( \sin(180^{\circ}+\theta)), the value is (-0.6000). For ( \cos155^{\circ}), the value is (-0.9063).