Precalculus by Richard Wright

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# 5-02 Fundamental Trigonometric Identities Part B

Summary: In this section, you will:

• Factor and multiply trigonometric expressions.
• Use trigonometric identities with rational expressions.
• Use trigonometric substitution.

SDA NAD Content Standards (2018): PC.5.1

The coefficient of friction is a measure of how slippery a surface is and is used to calculate force of friction. The fundamental trigonometric identities can be used to simplify expressions for the coefficient of friction.

## More Applications of the Fundamental Trigonometric Identities

Review the fundamental trigonometric identities in lesson 5-01. This lesson will continue using the identities to simplify trigonometric expressions.

#### Example 1: Factor Trigonometric Expressions

Factor the expression, then use fundamental trigonometric identities to simplify.

$$\csc^4 x - \cot^4 x$$

###### Solution

This is a difference of squares. $$\left(a^2 – b^2\right) = \left(a - b\right)\left(a + b\right)$$

$$\csc^4 x - \cot^4 x$$

$$\left(\csc^2 x - \cot^2 x\right)\left(\csc^2 x + \cot^2 x\right)$$

The Pythagorean identity, 1 + cot2 x = csc2 x, lets us substitute for csc2 x.

$$\left(\left(1 + \cot^2 x\right) - \cot^2 x\right) \left(\left(1 + \cot^2 x\right) + \cot^2 x\right)$$

$$1 + 2\cot^2 x$$

##### Try It 1

Factor and simplify $$1 + \cos x - \cos^2 x - \cos^3 x$$.

$$\sin^2 x\left(1 + \cos x\right)$$

#### Example 2: Multiply Trigonometric Expressions

Multiply and use trigonometric identities to simplify.

$$\left(3 \sin x - 3\right)\left(3 \sin x + 3\right)$$

###### Solution

Multiply.

$$\left(3 \sin x - 3\right)\left(3 \sin x + 3\right) = 9 \sin^{2} x - 9$$

Factor out the 9.

$$9\left(\sin^{2} x - 1\right)$$

A Pythagorean identity relates sin2 x and 1. Solve $$\sin^{2} x + \cos^{2} x = 1$$ for sin2 x to get $$\sin^{2} x = 1 - \cos^{2} x$$ and substitute.

$$9\left(1 - \cos^{2} x - 1\right)$$

$$9\left(-\cos^{2} x\right)$$

$$–9 \cos^{2} x$$

##### Try It 2

Simplify $$\left(\cos x - 1\right)\left(\cos x + 1\right)$$.

$$-\sin^2 x$$

#### Example 3: Add or Subtract Fractions

Add or subtract the expressions and use trigonometric identities to simplify.

$$\frac{1}{\cos x + 1} - \frac{1}{\cos x - 1}$$

###### Solution

Use a common denominator to add the fractions.

$$\frac{1}{\cos x + 1} - \frac{1}{\cos x - 1}$$

$$\frac{\cos x - 1}{\left(\cos x + 1\right)\left(\cos x - 1\right)} - \frac{\cos x + 1}{\left(\cos x + 1\right)\left(\cos x - 1\right)}$$

$$\frac{–2}{\cos^{2} x - 1}$$

Use the Pythagorean identity, $$\sin^{2} x + \cos^{2} x = 1$$, solved for cos2 x to substitute. $$\cos^{2} x = 1 - \sin^{2} x$$

$$\frac{–2}{1 - \sin^{2} x - 1}$$

$$\frac{–2}{-\sin^{2} x}$$

$$2 \csc^{2} x$$

##### Try It 3

Simplify $$\frac{1}{1 + \sec x} - \frac{1}{1 - \sec x}$$

−2 csc x cot x

#### Example 4: Rewrite Fractions

Rewrite the fraction so it is not in fractional form.

$$\frac{1}{\csc x - \cot x}$$

###### Solution

Multiply the numerator and denominator by the conjugate of the denominator. This will make the denominator have csc2 x and cot2 x which are related by a Pythagorean identity.

$$\frac{1}{\csc x - \cot x}$$

$$\frac{\csc x + \cot x}{\left(\csc x - \cot x\right)\left(\csc x + \cot x\right)}$$

$$\frac{\csc x + \cot x}{\csc^{2} x - \cot^{2} x}$$

Substitute the Pythagorean identity, $$1 + \cot^{2} x = \csc^{2} x$$.

$$\frac{\csc x + \cot x}{\left(1 + \cot^{2} x\right) - \cot^{2} x}$$

$$\csc x + \cot x$$

##### Try It 4

Rewrite the fraction so that it is not in fractional form: $$\frac{\sin^2 x}{1 - \cos x}$$.

1 + cos x

#### Example 5: Use Trigonometric Substitution

Use trigonometric substitution to write the algebraic expression as a trigonometric function.

$$\sqrt{16 - x^{2}}$$; x = 4 sin θ

###### Solution

Substitute the trigonometric expression for x.

$$\sqrt{16 - x^{2}} = \sqrt{16 - \left(4 \sin θ\right)^{2}}$$

$$\sqrt{16 - 16 \sin^{2} θ}$$

$$\sqrt{16\left(1 - \sin^{2} θ\right)}$$

The Pythagorean identity, $$\sin^{2} x + \cos^{2} x = 1$$, relates sin2 x and 1. Solve this for sin2 x and substitute. $$\sin^{2} x = 1 - \cos^{2} x$$

$$\sqrt{16\left(1 - \left(1 - \cos^{2} θ \right)\right)}$$

$$\sqrt{16 \cos^{2} θ }$$

$$4 \cos θ$$

##### Try It 5

Use trigonometric substitution to write the algebraic expression as a trigonometric function.

$$\sqrt{4 + x^2}; x = 2\cot α$$

2 csc α

##### Lesson Summary

###### Fundamental Trigonometric Identities

Reciprocal Identities

 $$\sin u = \frac{1}{\csc u}$$ $$\csc u = \frac{1}{\sin u}$$ $$\cos u = \frac{1}{\sec u}$$ $$\sec u = \frac{1}{\cos u}$$ $$\tan u = \frac{1}{\cot u}$$ $$\cot u = \frac{1}{\tan u}$$

Quotient Identities

 $$\tan u = \frac{\sin u}{\cos u}$$ $$\cot u = \frac{\cos u}{\sin u}$$

Pythagorean Identities

 $$\sin^{2} u + \cos^{2} u = 1$$ $$\tan^{2} u + 1 = \sec^{2} u$$ $$1 + \cot^{2} u = \csc^{2} u$$

Even/Odd Identities

 Even cos(–u) = cos u sec(–u) = sec u Odd sin(–u) = –sin u csc(–u) = –csc u tan(–u) = –tan u cot(–u) = –cot u

Cofunction Identities

 $$\sin \left(\frac{π}{2} - u\right) = \cos u$$ $$\cos \left(\frac{π}{2} - u\right) = \sin u$$ $$\tan \left(\frac{π}{2} - u\right) = \cot u$$ $$\cot \left(\frac{π}{2} - u\right) = \tan u$$ $$\sec \left(\frac{π}{2} - u\right) = \csc u$$ $$\csc \left(\frac{π}{2} - u\right) = \sec u$$

## Practice Exercises

Factor the expression, then use fundamental trigonometric identities to simplify.

1. sin2 x csc2 x − sin2 x
2. cot4 x + 2cot2 x + 1
3. tan4 x − sec4 x
4. Multiply and use trigonometric identities to simplify.

5. (cos x + sin x)2
6. (2sec x + 2)(2sec x − 2)
7. (csc x − cot x)(csc x + cot x)
8. Add or subtract the expressions and use trigonometric identities to simplify.

9. $$\frac{1}{1 + \sin x} + \frac{1}{1 - \sin x}$$
10. $$\frac{\cos x}{1 - \sin x} - \frac{\sin x}{\cos x}$$
11. $$\cot x - \frac{\csc^2 x}{\cot x}$$
12. Rewrite the fraction so it is not in fractional form.

13. $$\frac{\cos^2 x}{1 - \sin x}$$
14. $$\frac{4}{\sec x + \tan x}$$
15. Use trigonometric substitution to write the algebraic expression as a trigonometric expression.

16. $$\sqrt{4 - x^2}; x = 2\sin θ$$
17. $$\sqrt{x^2 - 36}; x = 6\csc θ$$
18. Rewrite the expression as a single logarithm.

19. $$\ln |\sin x| + \ln |\csc x|$$
20. Problem Solving

21. If an object slides down an inclined surface at a constant speed, the force of friction has to equal the component of the object's weight pulling it down the surface. The equation becomes

$$μW\cos θ = W \sin θ$$

where μ is the coefficient of friction, W is the weight, and θ is the incline of the surface. Solve the equation for μ and use fundamental trigonometric identities to simplify the expression.

22. Mixed Review

23. (5-01) Simplify $$\left(1 - \cos^2 x\right) \csc x$$.
24. (5-01) Given $$\sin θ = -\frac{\sqrt{5}}{5}$$ and $$\tan θ < 0$$, evaluate cos θ and cot θ.
25. (4-05) Evaluate $$\tan \frac{7π}{4}$$ using reference angles.
26. (3-05) A substance has a half-life of 5.00 minutes. If the initial amount of the substance was 20.0 grams, how many half-lives will have passed before the substance decays to 13.0 grams?
27. (2-06) Find all the solutions of $$0 = x^3 - 7x^2 + 17x - 15$$.

1. cos2 x
2. csc4 x
3. -2 tan2 x - 1
4. 2 cos x sin x + 1
5. 4 tan2 x
6. 1
7. 2 sec2 x
8. sec x
9. −tan x
10. 1 + sin x
11. 4 sec x − 4 tan x
12. 2 cos θ
13. 6 cot θ
14. 0
15. μ = tan θ
16. sin x
17. $$\cos θ = \frac{2\sqrt{5}}{5}$$, cot θ = −2
18. −1
19. 0.621
20. 3, 2 + i, 2 − i