Precalculus by Richard Wright
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
Summary: In this section, you will:
SDA NAD Content Standards (2018): PC.5.1
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
$$\tan θ = \frac{v^2}{gr}$$
There are four main uses of 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.
Try
If \(\cos α = \frac{4}{5}\) and tan α < 0, evaluate a) sin α and b) cot α.
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).
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, \(\sin α = -\frac{3}{5}\).
A quotient identity relates cotangent with cosine and sine.
\(\cot α = \frac{\cos α}{\sin α}\) |
\(\cot α = \frac{\frac{4}{5}}{-\frac{3}{5}}\) |
\(\cot α = -\frac{4}{3}\) |
If \(\tan θ = -\frac{5}{6}\) and \(\sin θ = \frac{\sqrt{61}}{5}\), find a) cos θ and b) csc θ.
Because tangent < 0 and sine > 0, angle θ is in quadrant II and the signs of the trigonometric functions should be for that quadrant.
A quotient identity relates tan θ, sin θ, and cos θ.
\(\tan θ = \frac{\sin θ}{\cos θ}\) |
\(-\frac{5}{6} = \frac{\frac{\sqrt{61}}{5}}{\cos θ}\) |
\(\cos θ = \frac{\frac{\sqrt{61}}{5}}{-\frac{5}{6}}\) |
\(\cos θ = -\frac{6\sqrt{61}}{25}\) |
A reciprocal identity relates sin θ and csc θ
\(\csc θ = \frac{1}{\sin θ}\) |
\(\csc θ = \frac{1}{\frac{\sqrt{61}}{5}}\) |
\(\csc θ = \frac{5\sqrt{61}}{61}\) |
If \(\sin θ = -\frac{24}{25}\) and cos θ < 0, find a) cos θ and b) cot θ.
\(-\frac{7}{25}, \frac{7}{24}\)
Simplify \(\cos x \tan^{2} x + \cos x\).
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 sec^{2} x for \(\left(\tan^2 x + 1\right)\).
$$\cos x \sec^2 x$$
A reciprocal identity allows sec x to be written as \(\frac{1}{\cos x}\).
\(\cos x \frac{1}{\cos^2 x}\) |
\(\frac{1}{\cos x}\) |
\(\sec x\) |
Simplify \(\cos\left(\frac{π}{2} - x\right) \csc\left(–x\right)\).
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
Simplify \(\sec x - \sec x \sin^2 x\).
cos x
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
\(\sin \left(\frac{π}{2} - u\right) = \cos u\) | \(\cos \left(\frac{π}{2} - u\right) = \sin u\) |
\(\tan \left(\frac{π}{2} - u\right) = \cot u\) | \(\cot \left(\frac{π}{2} - u\right) = \tan u\) |
\(\sec \left(\frac{π}{2} - u\right) = \csc u\) | \(\csc \left(\frac{π}{2} - u\right) = \sec u\) |
Try
Helpful videos about this lesson.
Use the given values to evaluate all six trigonometric functions.
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 |
Use the fundamental identities to simplify the expression. There may be more than one correct answer.
Mixed Review