91Ó°ÊÓ

Verify each identity. \(\sec x-\sec x \sin ^{2} x=\cos x\)

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$ \sin \frac{5 \pi}{12} \cos \frac{\pi}{4}-\cos \frac{5 \pi}{12} \sin \frac{\pi}{4} $$

Use a half-angle formula to find the exact value of each expression. $$ \tan 112.5^{\circ} $$

Verify each identity. \(\left(\cot ^{2} \theta+1\right)\left(\sin ^{2} \theta+1\right)=\cot ^{2} \theta+2\)

Throwing events in track and field include the shot put, the discus throw, the hammer throw, and the javelin throw. The distance that the athlete can achieve depends on the initial speed of the object thrown and the angle above the horizontal at which the object leaves the hand. This angle is represented by \(\theta\) in the figure shown. The distance, \(d,\) in feet, that the athlete throws is modeled by the formula $$ d=\frac{v_{0}^{2}}{16} \sin \theta \cos \theta $$ in which \(v_{0}\) is the initial speed of the object thrown, in feet per second, and \(\theta\) is the angle, in degrees, at which the object leaves the hand. a. Use an identity to express the formula so that it contains the sine function only. b. Use your formula from part (a) to find the angle, \(\theta,\) that produces the maximum distance, \(d,\) for a given initial speed, \(v_{0}\).

Use this information to solve: The speed of a supersonic aircraft is usually represented by a Mach number, named after Austrian physicist Ernst Mach \((1838-1916) .\) A Mach number is the speed of the aircraft, in miles per hour, divided by the speed of sound, approximately 740 miles per hour. Thus, a plane flying at twice the speed of sound has a speed, M, of Mach 2. (GRAPH CANNOT COPY). If an aircraft has a speed greater than Mach 1 , a sonic boom is heard, created by sound waves that form a cone with a vertex angle \(\theta,\) shown in the figure. The relationship between the cone's vertex angle, \(\theta,\) and the Mach speed, M, of an aircraft that is flying faster than the speed of sound is given by $$ \sin \frac{\theta}{2}=\frac{1}{M} $$ If \(\theta=\frac{\pi}{6},\) determine the Mach speed, \(M,\) of the aircraft. Express the speed as an exact value and as a decimal to the nearest tenth.

Use words to describe the formula for: the cosine of double an angle. (Describe one of the three formulas.)

Graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, this indicates that the equation is not an identity. In these exercises, find a value of \(x\) for which both sides are defined but not equal. $$ \cos \left(\frac{3 \pi}{2}-x\right)=-\sin x $$

Without showing algebraic details, describe in words how to reduce the power of \(\cos ^{4} x\).

Write each trigonometric expression as an algebraic expression (that is, without any trigonometric functions). Assume that x and y are positive and in the domain of the given inverse trigonometric function. $$ \sin \left(\tan ^{-1} x-\sin ^{-1} y\right) $$