<meta http-equiv="refresh" content="1; url=/nojavascript/">
Dismiss
Skip Navigation
You are viewing an older version of this Concept. Go to the latest version.

Simplifying Trigonometric Expressions using Sum and Difference Formulas

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

Atoms Practice
0%
Progress
Practice Simplifying Trigonometric Expressions using Sum and Difference Formulas
Practice
Progress
0%
Practice Now
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) .

Answers

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)

Image Attributions

Reviews

Please wait...
Please wait...

Original text