91Ó°ÊÓ

Use the sum-to-product formulas to write the sum or difference as a product. $$\sin 5 \theta-\sin 3 \theta$$

A batted baseball leaves the bat at an angle of \(\theta\) with the horizontal and an initial velocity of \(v_{0}=100\) feet per second. The ball is caught by an outfielder 300 feet from home plate (see figure). Find \(\theta\) if the range \(r\) of a projectile is given by \(r=\frac{1}{32} v_{0}^{2} \sin 2 \theta\).

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\boldsymbol{\theta},\) where \(0<\boldsymbol{\theta}<\pi / 2\) $$\sqrt{9-x^{2}}, x=3 \cos \theta$$

Determine whether the statement is true or false. Justify your answer. $$\tan \left(x-\frac{\pi}{4}\right)=\frac{\tan x+1}{1-\tan x}$$

Use the sum-to-product formulas to write the sum or difference as a product. $$\cos 6 x+\cos 2 x$$

a sharpshooter intends to hit a target at a distance of 1000 yards with a gun that has a muzzle velocity of 1200 feet per second (see figure). Neglecting air resistance, determine the gun's minimum angle of elevation \(\theta\) if the range \(r\) is given by $$r=\frac{1}{32} v_{0}^{2} \sin 2 \theta$$

Determine whether the statement is true or false. Justify your answer. $$\sin \left(x-\frac{\pi}{2}\right)=-\cos x$$

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\boldsymbol{\theta},\) where \(0<\boldsymbol{\theta}<\pi / 2\) $$\sqrt{64-16 x^{2}}, \quad x=2 \cos \theta$$

Use the sum-to-product formulas to write the sum or difference as a product. $$\cos x+\cos 4 x$$

Verify the identity. $$\cos (n \pi+\theta)=(-1)^{n} \cos \theta, n \text { is an integer }$$