Precalculus by Richard Wright

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Start children off on the way they should go, and even when they are old they will not turn from it. Proverbs‬ ‭22‬:‭6‬ ‭NIV‬‬

5-01 Fundamental Trigonometric Identities Part A

Mr. Wright teaches the lesson.

Summary: In this section, you will:

SDA NAD Content Standards (2018): PC.5.1

banked curve
Figure 1: Banked corner on Autodromo nazionale di Monza. credit (wikicommons/public domain)

Trigonometric identities can be used to simplify expressions. For examples, figure 1 shows a banked curve of a race track. Banked curves are designed to eliminate the need for friction to turn the car at a given speed. The equations modeling the banked curve are

$$F_N \sin θ = \frac{mv^{2}}{r}$$
\(F_N \cos θ = mg\)

Trigonometric identities simplify these equations into

$$\tan θ = \frac{v^2}{gr}$$

Fundamental Trigonometric Identities

There are four main uses of trigonometric identities.

  1. Evaluate trigonometric expressions
  2. Simplify trigonometric expressions
  3. Create additional trigonometric identities
  4. Solve trigonometric equations
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

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

To evaluate trigonometric expressions using identities, replace a complicated expression with a simpler one. If there are more than one trigonometric functions in the expression, try to replace them with a single function. Repeat this until the functions are simple to evaluate.

Evaluate Trigonometric Functions Using Identities

Try

Example 1: Evaluate a Trigonometric Function

If \(\cos α = \frac{4}{5}\) and tan α < 0, evaluate a) sin α and b) cot α.

Solution

Because cosine > 0 and tangent < 0, angle α is in quadrant IV and the signs of the trigonometric functions should be for that quadrant (see Lesson 4-05).

  1. An identity relating cos α and sin α is a Pythagorean Identity.

    \(\sin^2 α + \cos^2 α = 1\)
    \(\sin^2 α + \left(\frac{4}{5}\right)^2 = 1\)
    \(\sin^2 α = \frac{9}{25}\)
    \(\sin α = ±\frac{3}{5}\)

    Since sine is negative in quadrant IV, \(\sin α = -\frac{3}{5}\).

  2. A quotient identity relates cotangent with cosine and sine.

    \(\cot α = \frac{\cos α}{\sin α}\)
    \(\cot α = \frac{\frac{4}{5}}{-\frac{3}{5}}\)
    \(\cot α = -\frac{4}{3}\)

Example 2: Evaluate a Trigonometric Function

If \(\tan θ = -\frac{5}{6}\) and \(\sin θ = \frac{\sqrt{61}}{5}\), find a) cos θ and b) csc θ.

Solution

Because tangent < 0 and sine > 0, angle θ is in quadrant II and the signs of the trigonometric functions should be for that quadrant.

  1. A quotient identity relates tan θ, sin θ, and cos θ.

    \(\tan θ = \frac{\sin θ}{\cos θ}\)
    \(-\frac{5}{6} = \frac{\frac{\sqrt{61}}{5}}{\cos θ}\)
    \(\cos θ = \frac{\frac{\sqrt{61}}{5}}{-\frac{5}{6}}\)
    \(\cos θ = -\frac{6\sqrt{61}}{25}\)
  2. A reciprocal identity relates sin θ and csc θ

    \(\csc θ = \frac{1}{\sin θ}\)
    \(\csc θ = \frac{1}{\frac{\sqrt{61}}{5}}\)
    \(\csc θ = \frac{5\sqrt{61}}{61}\)
Try It 1

If \(\sin θ = -\frac{24}{25}\) and cos θ < 0, find a) cos θ and b) cot θ.

Answer

\(-\frac{7}{25}, \frac{7}{24}\)

Example 3: Simplify a Trigonometric Expression

Simplify \(\cos x \tan^{2} x + \cos x\).

Solution

Notice that there is a cos x each term. Factor out the cos x.

\(\cos x \tan^2 x + \cos x\)
\(\left(\cos x\right)\left(\tan^2 x + 1\right)\)

A Pythagorean identity is \(\tan^2 u + 1 = \sec^2 u\), so substitute sec2 x for \(\left(\tan^2 x + 1\right)\).

$$\cos x \sec^2 x$$

A reciprocal identity allows sec x to be written as \(\frac{1}{\cos x}\).

\(\cos x \frac{1}{\cos^2 x}\)
\(\frac{1}{\cos x}\)
\(\sec x\)

Example 4: Simplify a Trigonometric Expression

Simplify \(\cos\left(\frac{π}{2} - x\right) \csc\left(–x\right)\).

Solution

A cofunction identity says \(\cos\left(\frac{π}{2} - x\right) = \sin x\).

\(\cos\left(\frac{π}{2} - x\right)\csc\left(–x\right)\)
\(\sin x \csc\left(–x\right)\)

An even/odd identity says \(\csc\left(–x\right) = -\csc x\).

$$\sin x \left(-\csc x\right)$$

A reciprocal identity says \(\csc x = \frac{1}{\sin x}\).

$$\sin x \left(-\frac{1}{\sin x}\right)$$
–1

Try It 2

Simplify \(\sec x - \sec x \sin^2 x\).

Answers

cos x

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

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

Evaluate Trigonometric Functions Using Identities

Try

Helpful videos about this lesson.

Practice Exercises

    Use the given values to evaluate all six trigonometric functions.

  1. \(\sin ϕ = \frac{2}{3}, \cos ϕ = \frac{\sqrt{5}}{3}\)
  2. \(\tan θ = -\frac{7}{24}, \sec θ = -\frac{25}{24}\)
  3. \(\csc α = -\frac{6}{5}, \cot α = \frac{\sqrt{11}}{5}\)
  4. \(\sin x = \frac{5}{13}, \tan x < 0\)
  5. \(\tan y = undefined, \sin y < 0\)
  6. Match the trigonometric expression with one of the following.

    (a) \(2\sin^2 x - 1\) (b) sin x
    (c) sec x (d) cos x
    (e) csc x (f) tan x
  7. \(\cos x \tan x\)
  8. \(\left(\sec x\right)\left(1 - \sin^2 x\right)\)
  9. \(\sin^4 x - \cos^4 x\)
  10. \(-\tan\left(-x\right)\sec\left(\frac{π}{2} - x\right)\)
  11. \(\sin x + \sin x \cot^2 x\)
  12. Use the fundamental identities to simplify the expression. There may be more than one correct answer.

  13. \(\tan x \sin \left(\frac{π}{2} - x\right) + \cot x \sin^2 x \sec x\)
  14. \(\tan α\left(\cot α + \tan α\right)\)
  15. \(\frac{\sin^2 θ}{1 - \cos θ}\)
  16. \(\frac{1}{1 + \cot^2 ϕ}\)
  17. \(\frac{1 - \sin^2 x}{\sec x}\)
  18. Mixed Review

  19. (4-11) A ship leaves port and travels for 2 hours at 1.5 knots due south. Then it changes course due west for 1 hour. Find the distance and bearing from the starting point.
  20. (4-10) A park is in the shape of a right triangle with the perpendicular side lengths 400 ft and 500 ft. What is the size of the acute angle adjacent to the 400 ft side (round to the nearest tenth)?
  21. (3-02) Rewrite the logarithm in exponential form: \(\log_3 81 = 4\).
  22. (2-01) Divide \(\frac{2 + i}{1 - i}\).
  23. (1-08) If \(f(x) = x^2 + 1\) and \(g(x) = x - 4\), find \(\left(f ∘ g\right)(x)\).

Answers

  1. \(\tan ϕ = \frac{2\sqrt{5}}{5}\), \(\csc ϕ = \frac{3}{2}\), \(\sec ϕ = \frac{3\sqrt{5}}{5}\), \(\cot ϕ = \frac{\sqrt{5}}{2}\)
  2. \(\sin θ = \frac{7}{25}\), \(\cos θ = -\frac{24}{25}\), \(\csc θ = \frac{25}{7}\), \(\cot θ = -\frac{24}{7}\)
  3. \(\sin α = -\frac{5}{6}\), \(\cos α = -\frac{\sqrt{11}}{6}\), \(\tan α = \frac{5\sqrt{11}}{11}\), \(\sec α = -\frac{6\sqrt{11}}{11}\)
  4. \(\cos x = -\frac{12}{13}\), \(\tan x = -\frac{5}{12}\), \(\csc x = \frac{13}{5}\), \(\sec x = -\frac{13}{12}\), \(\cot x = -\frac{12}{5}\)
  5. sin y = −1, cos y = 0, csc y = −1, sec y = undefined, cot y = 0
  6. b
  7. d
  8. a
  9. c
  10. e
  11. 2 sin x
  12. sec2 α
  13. 1 + cos θ
  14. sin2 ϕ
  15. cos3 x
  16. 3.35 nautical miles at S 26.6° W
  17. 51.3°
  18. 34 = 81
  19. \(\frac{1 + 3i}{2}\)
  20. x2 − 8x + 17