Chapter Questions
Derive the difference formula for cosines.
Derive(a) the sum formula for cosines; (b) the cofunction formulas for sines and cosines.
Derive the difference formula for sines.
Derive the difference formula for tangents.
Given $\sin x=\frac{3}{5}, x$ in quadrant $\mathrm{I}$, and $\cos y=\frac{2}{3}, y$ in quadrant IV, find (a) $\cos (x+y)$; (b) $\tan (x+y)$;(c) the quadrant in which $x+y$ must lie.
Derive the double-angle formulas for sine and cosine.
Given $\tan t=\frac{7}{24}, t$ in quadrant III, find $\sin 2 t$ and $\cos 2 t$.
Use a double-angle identity to derive an expression for $\cos 4 t$ in terms of $\cos t$.
Use a half-angle identity to derive an expression for $\cos ^4 t$ in terms of cosines with exponent 1.
Derive the half-angle formulas.
Given $\tan u=4, \pi<u<\frac{3 \pi}{2}$, find $\tan \frac{u}{2}$.
Derive the product-to-sum formulas.
Use a product-to-sum formula to rewrite $\cos 5 x \cos x$ as a sum.
Derive the sum-to-product formulas.
Use a sum-to-product formula to rewrite $\sin 5 u+\sin 3 u$ as a product.
Verify the identity $\sin 3 \theta=3 \sin \theta-4 \sin ^3 \theta$.
If $f(x)=\sin x$, show that the difference quotient for $f(x)$ can be written as $\sin x\left(\frac{\cos h-1}{h}\right)+\cos x \frac{\sin h}{h}$.
Verify the reduction formulas: (a) $\sin (\theta+\pi)=-\sin \theta$; (b) $\tan \left(\theta+\frac{\pi}{2}\right)=-\cot \theta$.
Find all solutions on the interval $[0,2 \pi)$ for $\cos t-\sin 2 t=0$.
Find all solutions on the interval $[0,2 \pi)$ for $\cos 5 x-\cos 3 x=0$.
Derive the sum formulas for sine and tangent.
Derive the cofunction formulas (a) for tangents and cotangents; (b) for secants and cosecants.
Use sum or difference formulas to find exact values for (a) $\sin \frac{5 \pi}{12}$; (b) $\cos 105^{\circ}$; (c) $\tan \left(-\frac{\pi}{12}\right)$.
Given $\sin u=-\frac{2}{5}, u$ in quadrant III, and $\cos v=\frac{3}{4}, v$ in quadrant IV, find (a) $\sin (u+v)$; (b) $\cos (u-v)$;(c) $\tan (v-u)$.
Derive (a) the double-angle formula for tangents; (b) the third form of the half-angle formula for tangents.
Given $\sec t=-3, \frac{\pi}{2}<t<\pi$, find (a) $\sin 2 t$; (b) $\tan 2 t$; (c) $\cos \frac{t}{2}$; (d) $\tan \frac{t}{2}$.
Use a double-angle identity to derive an expression for (a) $\sin 4 x$ in terms of $\sin x$ and $\cos x$; (b) $\cos 6 u$ in terms of $\cos u$.
Use a half-angle identity to derive an expression in terms of cosines with exponent 1 for (a) $\sin ^2 2 t \cos ^2 2 t$; (b) $\sin ^4 \frac{x}{2}$.
Complete the derivations of the product-to-sum and sum-to-product formulas (see Problems 24.13 and 24.14).
(a) Write $\sin 120 \pi t+\sin 110 \pi t$ as a product. (b) Write $\sin \frac{\pi n}{L} x \cos \frac{k \pi n}{L} t$ as a sum.
Verify that the following are identities: (a) $\frac{1+\sin 2 x-\cos 2 x}{1+\sin 2 x+\cos 2 x}=\tan x$; (b) $\tan \frac{u}{2}=\csc u-\cot u$;(c) $1+\tan \alpha \tan \frac{\alpha}{2}=\sec \alpha$; (d) $\frac{\cos a-\cos b}{\sin a-\sin b}=-\tan \left(\frac{a+b}{2}\right)$
Verify the reduction formulas:(a) $\sin (n \pi+\theta)=(-1)^n \sin \theta$, for $n$ any integer; (b) $\cos (n \pi+\theta)=(-1)^n \cos \theta$, for $n$ any integer.
If $f(x)=\cos x$, show that the difference quotient for $f(x)$ can be written as$$\cos x\left(\frac{\cos h-1}{h}\right)-\sin x \frac{\sin h}{h} .$$
Find all solutions on the interval $[0,2 \pi)$ for the following equations:(a) $\sin 2 \theta-\sin \theta=0$; (b) $\cos x+\cos 3 x=\cos 2 x$.
Find approximate values for all solutions on the interval $\left[0^{\circ}, 360^{\circ}\right)$ for $\cos x=2 \cos 2 x$.