91Ó°ÊÓ

Solve the trigonometric equations exactly on the indicated interval, \(0 \leq x<2 \pi\). $$\sin ^{2} x-\cos (2 x)=-\frac{1}{4}$$

Show that the area of a circle with radius \(r=\sec x\) is equal to \(\pi+\pi(\tan x)^{2}\)

Evaluate each expression exactly. $$\cos \left[\tan ^{-1}\left(\frac{7}{24}\right)\right]$$

Use the half-angle identities to find the desired function values. $$\text { If } \sin x=-0.3 \text { and } \sec x<0, \text { find } \cot \left(\frac{x}{2}\right)$$

Show that the area of a triangle with base \(b=\cos x\) and height \(h=\sec x\) is equal to \(\frac{1}{2}\)

Solve the trigonometric equations exactly on the indicated interval, \(0 \leq x<2 \pi\). $$\sin ^{2} x-2 \sin x=0$$

Show that the difference quotient for \(f(x)=\cos x\) is \(-\sin x\left(\frac{\sin h}{h}\right)-\cos x\left(\frac{1-\cos h}{h}\right)\) Plot \(Y_{1}=-\sin x\left(\frac{\sin h}{h}\right)-\cos x\left(\frac{1-\cos h}{h}\right)\) for a. \(h=1\) b. \(h=0.1\) c. \(h=0.01\) What function does the difference quotient for \(f(x)=\cos x\) resemble when \(h\) approaches zero?

Solve the trigonometric equations exactly on the indicated interval, \(0 \leq x<2 \pi\). $$\cos ^{2} x+2 \sin x+2=0$$

Evaluate each expression exactly. $$\tan \left[\sin ^{-1}\left(\frac{3}{5}\right)\right]$$

Use the half-angle identities to find the desired function values. $$\text { If } \sec x=2.5 \text { and } \tan x>0, \text { find } \cot \left(\frac{x}{2}\right)$$