Precalculus Honors | Standards PC.PAFR.6.1 & 6.2
This presentation covers the manipulation and application of trigonometric identities and formulas as outlined in the South Carolina College- and Career-Ready Standards:
Reciprocal: \(\csc x = \frac{1}{\sin x} \quad \sec x = \frac{1}{\cos x} \quad \cot x = \frac{1}{\tan x}\)
Quotient: \(\tan x = \frac{\sin x}{\cos x} \quad \cot x = \frac{\cos x}{\sin x}\)
1. Simplify: \(\sin x \cdot \csc x\)
\(\sin x \cdot \left(\frac{1}{\sin x}\right)\) \(\rightarrow\) 1
2. Simplify: \(\frac{\sec \theta}{\csc \theta}\)
\(\frac{1/\cos \theta}{1/\sin \theta} = \frac{1}{\cos \theta} \cdot \frac{\sin \theta}{1}\) \(\rightarrow\) \(\tan \theta\)
3. Simplify: \(\cos \beta \cdot \sec \beta - \sin^2 \beta\)
\(1 - \sin^2 \beta\) \(\rightarrow\) \(\cos^2 \beta\)
1. Simplify: \(\cos x \cdot \tan x\)
\(\cos x \cdot \left(\frac{\sin x}{\cos x}\right)\) \(\rightarrow\) \(\sin x\)
2. Simplify: \(\frac{\sin \alpha}{\cos \alpha \cdot \tan \alpha}\)
\(\frac{\sin \alpha}{\cos \alpha \cdot (\sin \alpha / \cos \alpha)} = \frac{\sin \alpha}{\sin \alpha}\) \(\rightarrow\) 1
3. Simplify: \(\cot \theta \cdot \sin \theta\)
\(\left(\frac{\cos \theta}{\sin \theta}\right) \cdot \sin \theta\) \(\rightarrow\) \(\cos \theta\)
\(\sin^2 x + \cos^2 x = 1\)
\(1 + \tan^2 x = \sec^2 x\)
\(1 + \cot^2 x = \csc^2 x\)
1. Simplify: \((1 - \sin^2 x)(\sec^2 x)\)
\(\cos^2 x \cdot \frac{1}{\cos^2 x}\) \(\rightarrow\) 1
2. Simplify: \(\tan^2 \theta - \sec^2 \theta\)
\(\tan^2 \theta - (1 + \tan^2 \theta)\) \(\rightarrow\) -1
3. Verify: \(\sin^2 \alpha \cdot \cot^2 \alpha + \sin^2 \alpha = 1\)
\(\sin^2 \alpha (\cot^2 \alpha + 1) = \sin^2 \alpha \cdot \csc^2 \alpha\) \(\rightarrow\) 1 = 1 ✓
Even/Odd: \(\sin(-x) = -\sin x, \quad \cos(-x) = \cos x, \quad \tan(-x) = -\tan x\)
Cofunction: \(\sin(\frac{\pi}{2} - x) = \cos x, \quad \tan(\frac{\pi}{2} - x) = \cot x, \quad \sec(\frac{\pi}{2} - x) = \csc x\)
1. Simplify: \(\sin(-x) \cdot \csc x\)
\(-\sin x \cdot \frac{1}{\sin x}\) \(\rightarrow\) -1
2. Simplify: \(\cos(-x) + \sin(-x)\tan x\)
\(\cos x - \sin x(\frac{\sin x}{\cos x}) = \frac{\cos^2 x - \sin^2 x}{\cos x}\) \(\rightarrow\) \(\frac{\cos 2x}{\cos x}\)
3. Simplify: \(\frac{\tan(-x)}{\sin(-x)}\)
\(\frac{-\tan x}{-\sin x} = \frac{\sin x / \cos x}{\sin x}\) \(\rightarrow\) \(\sec x\)
1. Simplify: \(\cos(\frac{\pi}{2} - x) \cdot \csc x\)
\(\sin x \cdot \frac{1}{\sin x}\) \(\rightarrow\) 1
2. Simplify: \(\frac{\sin(\frac{\pi}{2} - x)}{\cos(\frac{\pi}{2} - x)}\)
\(\frac{\cos x}{\sin x}\) \(\rightarrow\) \(\cot x\)
3. Simplify: \(\tan(\frac{\pi}{2} - \theta) \cdot \tan \theta\)
\(\cot \theta \cdot \tan \theta = \frac{1}{\tan \theta} \cdot \tan \theta\) \(\rightarrow\) 1
| Function | Sum Formula |
|---|---|
| Sine | \(\sin(u+v) = \sin u \cos v + \cos u \sin v\) |
| Cosine | \(\cos(u+v) = \cos u \cos v - \sin u \sin v\) |
| Tangent | \(\tan(u+v) = \frac{\tan u + \tan v}{1 - \tan u \tan v}\) |
| Function | Difference Formula |
|---|---|
| Sine | \(\sin(u-v) = \sin u \cos v - \cos u \sin v\) |
| Cosine | \(\cos(u-v) = \cos u \cos v + \sin u \sin v\) |
| Tangent | \(\tan(u-v) = \frac{\tan u - \tan v}{1 + \tan u \tan v}\) |
1. Exact value: \(\sin(75^\circ)\) using \(45^\circ + 30^\circ\)
\(\sin 45 \cos 30 + \cos 45 \sin 30 = \frac{\sqrt{2}}{2}\frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2}\frac{1}{2}\) \(\rightarrow\) \(\frac{\sqrt{6}+\sqrt{2}}{4}\)
2. Exact value: \(\sin(15^\circ)\) using \(45^\circ - 30^\circ\)
\(\sin 45 \cos 30 - \cos 45 \sin 30 = \frac{\sqrt{2}}{2}\frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2}\frac{1}{2}\) \(\rightarrow\) \(\frac{\sqrt{6}-\sqrt{2}}{4}\)
3. Simplify: \(\sin(x + \pi)\)
\(\sin x \cos \pi + \cos x \sin \pi = \sin x(-1) + \cos x(0)\) \(\rightarrow\) \(-\sin x\)
1. Exact value: \(\cos(105^\circ)\) using \(60^\circ + 45^\circ\)
\(\cos 60 \cos 45 - \sin 60 \sin 45 = \frac{1}{2}\frac{\sqrt{2}}{2} - \frac{\sqrt{3}}{2}\frac{\sqrt{2}}{2}\) \(\rightarrow\) \(\frac{\sqrt{2}-\sqrt{6}}{4}\)
2. Exact value: \(\cos(\frac{\pi}{12})\) using \(\frac{\pi}{3} - \frac{\pi}{4}\)
\(\cos \frac{\pi}{3} \cos \frac{\pi}{4} + \sin \frac{\pi}{3} \sin \frac{\pi}{4} = \frac{1}{2}\frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2}\frac{\sqrt{2}}{2}\) \(\rightarrow\) \(\frac{\sqrt{2}+\sqrt{6}}{4}\)
3. Simplify: \(\cos(x - \frac{\pi}{2})\)
\(\cos x \cos \frac{\pi}{2} + \sin x \sin \frac{\pi}{2} = \cos x(0) + \sin x(1)\) \(\rightarrow\) \(\sin x\)
1. Exact value: \(\tan(75^\circ)\) using \(45^\circ + 30^\circ\)
\(\frac{\tan 45 + \tan 30}{1 - \tan 45 \tan 30} = \frac{1 + \sqrt{3}/3}{1 - (1)(\sqrt{3}/3)}\) \(\rightarrow\) \(2 + \sqrt{3}\)
2. Exact value: \(\tan(15^\circ)\) using \(45^\circ - 30^\circ\)
\(\frac{\tan 45 - \tan 30}{1 + \tan 45 \tan 30} = \frac{1 - \sqrt{3}/3}{1 + (1)(\sqrt{3}/3)}\) \(\rightarrow\) \(2 - \sqrt{3}\)
3. Find \(\tan(u+v)\) if \(\tan u = 3\) and \(\tan v = 2\)
\(\frac{3+2}{1-(3)(2)} = \frac{5}{-5}\) \(\rightarrow\) -1
| Function | Double-Angle Formulas |
|---|---|
| Sine | \(\sin 2u = 2 \sin u \cos u\) |
| Cosine | \(\cos 2u = \cos^2 u - \sin^2 u\) |
| Cosine | \(\cos 2u = 2\cos^2 u - 1\) |
| Cosine | \(\cos 2u = 1 - 2\sin^2 u\) |
| Tangent | \(\tan 2u = \frac{2 \tan u}{1 - \tan^2 u}\) |
1. If \(\sin u = \frac{3}{5}\) and \(u\) is in Q1, find \(\sin 2u\)
\(\cos u = \frac{4}{5} \rightarrow 2(\frac{3}{5})(\frac{4}{5})\) \(\rightarrow\) \(\frac{24}{25}\)
2. Simplify: \(4 \sin x \cos x\)
\(2(2 \sin x \cos x)\) \(\rightarrow\) \(2 \sin 2x\)
3. Solve on \([0, 2\pi)\): \(\sin 2x - \cos x = 0\)
\(2 \sin x \cos x - \cos x = 0 \rightarrow \cos x(2 \sin x - 1) = 0\) \(\rightarrow\) \(x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{\pi}{6}, \frac{5\pi}{6}\)
1. If \(\cos u = -\frac{2}{3}\), find \(\cos 2u\)
\(2(-\frac{2}{3})^2 - 1 = 2(\frac{4}{9}) - 1 = \frac{8}{9} - \frac{9}{9}\) \(\rightarrow\) \(-\frac{1}{9}\)
2. Simplify: \(1 - 2 \sin^2(15^\circ)\)
\(\cos(2 \cdot 15^\circ) = \cos 30^\circ\) \(\rightarrow\) \(\frac{\sqrt{3}}{2}\)
3. Verify: \(\frac{1 + \cos 2x}{2} = \cos^2 x\)
\(\frac{1 + (2\cos^2 x - 1)}{2} = \frac{2\cos^2 x}{2}\) \(\rightarrow\) \(\cos^2 x = \cos^2 x\) ✓
1. If \(\tan u = \frac{1}{2}\), find \(\tan 2u\)
\(\frac{2(1/2)}{1 - (1/2)^2} = \frac{1}{1 - 1/4} = \frac{1}{3/4}\) \(\rightarrow\) \(\frac{4}{3}\)
2. Simplify: \(\frac{2 \tan(22.5^\circ)}{1 - \tan^2(22.5^\circ)}\)
\(\tan(2 \cdot 22.5^\circ) = \tan 45^\circ\) \(\rightarrow\) 1
3. Find \(\tan 2x\) if \(\sin x = \frac{5}{13}\) in Q2
\(\cos x = -\frac{12}{13}, \tan x = -\frac{5}{12} \rightarrow \frac{2(-5/12)}{1-(-5/12)^2}\) \(\rightarrow\) \(-\frac{120}{119}\)
Sine: \(\sin \frac{u}{2} = \pm \sqrt{\frac{1 - \cos u}{2}}\)
Cosine: \(\cos \frac{u}{2} = \pm \sqrt{\frac{1 + \cos u}{2}}\)
Tangent: \(\tan \frac{u}{2} = \frac{1 - \cos u}{\sin u} = \frac{\sin u}{1 + \cos u}\)
(Sign depends on the quadrant of \(u/2\))
1. Exact value: \(\sin(22.5^\circ)\)
\(\sqrt{\frac{1 - \cos 45}{2}} = \sqrt{\frac{1 - \sqrt{2}/2}{2}}\) \(\rightarrow\) \(\frac{\sqrt{2-\sqrt{2}}}{2}\)
2. If \(\cos u = \frac{1}{4}\) and \(u\) is in Q4, find \(\sin \frac{u}{2}\)
\(u/2\) is in Q2, so sine is positive: \(\sqrt{\frac{1 - 1/4}{2}} = \sqrt{\frac{3/4}{2}}\) \(\rightarrow\) \(\frac{\sqrt{6}}{4}\)
3. Simplify: \(\sqrt{\frac{1 - \cos 80^\circ}{2}}\)
\(\sin(\frac{80^\circ}{2})\) \(\rightarrow\) \(\sin 40^\circ\)
1. Exact value: \(\cos(75^\circ)\) using half of \(150^\circ\)
\(\sqrt{\frac{1 + \cos 150}{2}} = \sqrt{\frac{1 - \sqrt{3}/2}{2}}\) \(\rightarrow\) \(\frac{\sqrt{2-\sqrt{3}}}{2}\)
2. If \(\cos u = -\frac{1}{2}\) and \(u\) is in Q3, find \(\cos \frac{u}{2}\)
\(u/2\) is in Q2, so cosine is negative: \(-\sqrt{\frac{1 + (-1/2)}{2}} = -\sqrt{1/4}\) \(\rightarrow\) \(-\frac{1}{2}\)
3. Simplify: \(2 \cos^2(\frac{x}{2}) - 1\)
\(\cos(2 \cdot \frac{x}{2})\) \(\rightarrow\) \(\cos x\)
1. Exact value: \(\tan(15^\circ)\)
\(\frac{1 - \cos 30}{\sin 30} = \frac{1 - \sqrt{3}/2}{1/2}\) \(\rightarrow\) \(2 - \sqrt{3}\)
2. If \(\sin u = \frac{4}{5}\) in Q1, find \(\tan \frac{u}{2}\)
\(\cos u = \frac{3}{5} \rightarrow \frac{1 - 3/5}{4/5} = \frac{2/5}{4/5}\) \(\rightarrow\) \(\frac{1}{2}\)
3. Simplify: \(\frac{\sin 2x}{1 + \cos 2x}\)
This is the half-angle form for \(\tan(\frac{2x}{2})\) \(\rightarrow\) \(\tan x\)