8.6E Exercises

Basic Simplification

Simplify the expressions as instructed.

1. FOIL and simplify: \( (1-\sin x)(1+\sin x) \)
2. FOIL and simplify: \( (\cos \theta + \sin \theta)^2 \)
3. Convert to sines and cosines and simplify: \( \frac{\sin x}{\tan x} \)
4. Convert to sines and cosines and simplify: \( \cos^2 \theta + \frac{1}{\csc^2 \theta} \)
5. Convert to sines and cosines and simplify: \( \sin x \tan x + \cos x\)
6. Convert to sines and cosines and simplify: \( \frac{\sin \theta}{\cot \theta} + \cos \theta \)
7. Show the steps simplifying \( \frac{\tan \theta}{\csc \theta} \cos \theta \) to \( \sin^2 \theta\).

Answer

1. \( 1 - \sin^2 x \)
2. \( 1 + 2 \cos \theta \sin \theta \)
3. \( \cos x\)
4. \( 1\)
5. \( \frac{1}{\cos x}\) or \( \sec x\)
6. \( \frac{1}{\cos \theta} \) or \( \sec \theta\)
7. \( \frac{\tan \theta}{\csc \theta} \cos \theta = \dfrac{ \frac{\sin \theta}{\cos\theta}}{\frac{1}{\sin\theta}}\cos \theta = \frac{\sin^2 \theta}{\cos \theta} \cos \theta = \sin^2 \theta \)

Manipulating and Simplifying Trig Expressions

1. Write the expression \( \dfrac{\sin x + x}{\cos x}\) as two terms involving \( \tan x\) and/or \(\sec x\).

2. Write the expression \( \dfrac{\sin \theta}{\cos \theta} + \dfrac{\cos \theta}{1+\sin \theta}\) as a single fraction term and then simplify as far as possible.

3. Write the expression \( \dfrac{1}{1-\sin x} - \dfrac{1}{1+\sin x} \) as a single simplified term involving \( \tan x \) and/or \(\sec x\).

4. Write the expression \( \csc \theta + \cot \theta\) as a single fraction term involving \(\sin \theta\) and/or \(\cos \theta\)

5. Substitute \( \cos \theta\) for \(x\) in the expression \( \sqrt{ x^2 - 1} \) and simplify using the Pythagorean Identity (assume \( \theta\) is an angle such that \( \sin \theta \geq 0\)).

Answer

1. \( \tan x + x \sec x \)
2. \( \sec \theta\)
3. \( 2 \tan x \sec x \)
4. \( \dfrac{1+\cos \theta}{\sin \theta}\)
5. \( \sqrt{ \cos^2 \theta - 1} = \sqrt{\sin^2 \theta} = \sin \theta \)

Using Sum/Difference Formulas

(Make sure to do all of these.)

1. Simplify \(\sin\left( x + \pi \right)\).

2. Simplify \( \cos \left( \frac{\pi}{2} - \theta \right) \). (This is called a cofunction identity and is demonstrating the periodic behavior of cosine.)

3. Find a triple-angle formula for \(\sin(3A)\) by writing \( 3A = 2A + A\) and using a sum formula and double angle formula.

4. Use a sum formula backwards to evaluate \( \sin 15^\circ \cos 30^\circ + \cos 15^\circ \sin 30^\circ \) even though \( 15^\circ\) isn't a "typical" angle we have memorized.

5. (This is an expression that could show up in Calc I.) Plug in \(f(x) = \sin x\) to the expression

\[ \frac{ f(x + h) - f(x)}{h} \notag \]

and simplify until it looks like this: \( \sin x \left( \dfrac{\cos h - 1}{h} \right) + \cos x \left( \dfrac{ \sin h}{h} \right) \).

Answer

1. \( \sin x \)
2. \( \cos x\)
3. \( 4\sin A \cos^2 A - \sin A \)
4. This is equivalent to \( \sin (15^\circ +30^\circ) = \sin( 45^\circ\) = \frac{1}{\sqrt{2}} \).
5. Plugged in, you should have \( \dfrac{ \sin(x+h) - \sin x}{h} \), and you use the sum formula on the \( \sin(x+h)\) term. Then algebraically manipulate by factoring, splitting fractions, etc...

Formulas for Lowing Powers

The results of these problems provide a formula you can use to get rid of a squared trig term and replace it with some trig term to the first power only.

1. Use a double-angle formula to find an expression equivalent to \( \sin^2 x\) that involves \( \cos 2x\).
2. Use a double-angle formula to find an expression equivalent to \( \cos^2 x\) that involves \( \cos 2x \).
3. Use the formulas you derived to evaluate \(\sin^2 \left(\frac{\pi}{12}\right)\) even though it's not a "typical" angle.

Answer

1. Using \( \cos 2x = 1 - 2\sin^2 x\), we get \( \sin^2 x = \frac{ 1-\cos 2x}{2} \).
2. \( \cos^2 x = \frac{ 1+ \cos 2x}{2}\)
3. \( \frac{1}{2} - \frac{\sqrt{3}}{4}\) or \( \frac{2 - \sqrt{3}}{4}\) if you prefer.

Product-Sum Formulas

The results of these problems provide a way to rewrite a product of trig functions as a sum.

1. Recall that two equations can be added to form a new, also true, equation. That is,
   \[ A = B \text{ and } C = D \quad \implies \quad A+C = B+D. \notag \]
   Use this fact to add the sum formula for sine to the difference formula for sine.
2. Simplify the result so that one side has a single term involving \( \sin A \cos B\). Fully isolate this term to find a formula
   \[ \sin A \cos B = ... \notag \]
3. Use this formula to evaluate \( \sin\left( \frac{\pi}{8} \right) \cos \left( \frac{\pi}{8} \right) \) even though \(\pi/8\) isn't a "typical" angle.

Answer

1. \( \sin(A+B) + \sin(A-B) = \sin A \cos B + \cos A \sin B + \sin A \cos B - \cos A \sin B \)
2. \( \sin A \cos B = \frac{1}{2}(\sin(A+B) + \sin(A-B)) \)
3. \( \frac{1}{2\sqrt{2}}\) or, rationalized, \( \frac{\sqrt{2}}{4} \).

Fill-in-the-Blanks Practice for Memorization

Fill in the blanks in the identities below.

1. \( \sin^2 x + [ \quad \quad] = 1 \).
2. \( \tan^2 x + 1 = [ \quad \quad]\).
3. \( [ \quad \quad] + \cot^2 x = \csc^2 x \).
4. \( \sin (A+B) = \sin A \cos B \: [\quad] \cos A \sin B \).
5. \( \sin (A- B) = \sin A \cos B \: [\quad] \cos A \sin B \).
6. \( [ \quad \quad ] = \cos A \cos B - \sin A \sin B \).
7. \( \cos (A-B) = [ \quad \quad ] + [\quad \quad] \).
8. \( [ \quad \quad ] = \cos^2 A - \sin^2 A \).
9. \( \sin 2A = [\quad \quad] \).

Answer

After trying from scratch, check your answers against the previous section. Rinse and repeat.