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Simplifying Trigonometric Expressions using Sum and Difference Formulas

Simplify sine, cosine, and tangent of angles that are added or subtracted.

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Practice Simplifying Trigonometric Expressions using Sum and Difference Formulas
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Simplifying Trig Expressions using Sum and Difference Formulas

As Agent Trigonometry you are given this clue: $\sin(\frac{\pi}{2} - x)$ . How could you simplify this expression to make solving your case easier?

Guidance

We can also use the sum and difference formulas to simplify trigonometric expressions.

Example A

The $\sin a = -\frac{3}{5}$ and $\cos b =\frac{12}{13}$ . $a$ is in the $3^{rd}$ quadrant and $b$ is in the $1^{st}$ . Find $\sin(a+b)$ .

Solution: First, we need to find $\cos a$ and $\sin b$ . Using the Pythagorean Theorem, missing lengths are 4 and 5, respectively. So, $\cos a=-\frac{4}{5}$ because it is in the $3^{rd}$ quadrant and $\sin b = \frac{5}{13}$ . Now, use the appropriate formulas.

$\sin (a+b) &=\sin a \cos b + \cos a \sin b \\&= -\frac{3}{5}\cdot \frac{12}{13}+-\frac{4}{5}\cdot \frac{5}{13} \\&= -\frac{56}{65}$

Example B

Using the information from Example A, find $\tan (a+b)$ .

Solution: From the cosine and sine of $a$ and $b$ , we know that $\tan a=\frac{3}{4}$ and $\tan b=\frac{5}{12}$ .

$\tan (a+b) &=\frac{\tan a +\tan b}{1-\tan a \tan b} \\&= \frac{\frac{3}{4}+\frac{5}{12}}{1-\frac{3}{4}\cdot\frac{5}{12}} \\&= \frac{\frac{14}{12}}{\frac{11}{16}}=\frac{56}{33}$

Example C

Simplify $\cos (\pi - x)$ .

Solution: Expand this using the difference formula and then simplify.

$\cos (\pi - x) &=\cos \pi \cos x +\sin \pi \sin x \\&=-1\cdot \cos x +0\cdot \sin x \\&=-\cos x$

Concept Problem Revisit You can expand the expression using the difference formula and then simplify.

$\sin(\frac{\pi}{2} - x)=\sin \frac{\pi}{2} \cos x - \cos \frac{\pi}{2} \sin x \\&=1\cdot \cos x - 0\cdot \sin x \\&=cos x$

Guided Practice

1. Using the information from Example A, find $\cos(a-b)$ .

2. Simplify $\tan (x+\pi)$ .

1. $\cos(a-b) &=\cos a \cos b + \sin a \sin b \\&=-\frac{4}{5}\cdot \frac{12}{13}+-\frac{3}{5}\cdot\frac{5}{13} \\&=-\frac{63}{65}$

2. $\tan (x+\pi)&=\frac{\tan x +\tan \pi}{1-\tan x \tan \pi} \\&=\frac{\tan x +0}{1-\tan 0} \\&=\tan x$

Practice

$\sin a =-\frac{8}{17}, \pi \le a < \frac{3\pi}{2}$ and $\sin b =-\frac{1}{2}, \frac{3\pi}{2}\le b <2\pi$ . Find the exact trig values of:

1. $\sin (a+b)$
2. $\cos (a+b)$
3. $\sin (a-b)$
4. $\tan (a+b)$
5. $\cos (a-b)$
6. $\tan (a-b)$

Simplify the following expressions.

1. $\sin (2\pi-x)$
2. $\sin \left(\frac{\pi}{2}+x\right)$
3. $\cos (x+\pi)$
4. $\cos \left(\frac{3\pi}{2}-x\right)$
5. $\tan(x+2\pi)$
6. $\tan(x-\pi)$
7. $\sin \left(\frac{\pi}{6}-x\right)$
8. $\tan \left(\frac{\pi}{4}+x\right)$
9. $\cos \left(x-\frac{\pi}{3}\right)$

Determine if the following trig statements are true or false.

1. $\sin(\pi - x)=\sin (x-\pi)$
2. $\cos(\pi - x)=\cos (x-\pi)$
3. $\tan(\pi - x)=\tan (x-\pi)$