Trigonometry · Handbook · Mastering AMC 10/12

5.1 Trigonometric Identities

Theorem 5.1.1 · Trigonometric Identities

In a right triangle with an acute angle $a$ :

\text{Sine: } \sin(a) = \frac{\text{Opposite}}{\text{Hypotenuse}}

\text{Cosine: } \cos(a) = \frac{\text{Adjacent}}{\text{Hypotenuse}}

\text{Tangent: } \tan(a) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin(a)}{\cos(a)}

Learn it:Theorem 42.1.1 · Trigonometric Identities

Book p. 113Checked numerically

Remark 5.1.2

To remember the relationships, just use the mnemonic SOH, CAH, TOA:

SOH = Sin is Opposite over Hypotenuse
CAH = Cos is Adjacent over Hypotenuse
TOA = Tan is Opposite over Adjacent

Learn it:Remark 42.1.2

Book p. 114

5.2 More Trigonometric Identities

Theorem 5.2.1 · More Trigonometric Identities

\text{Cosecant: } \csc(a) = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{1}{\sin(a)}

\text{Secant: } \sec(a) = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{1}{\cos(a)}

\text{Cotangent: } \cot(a) = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{1}{\tan(a)} = \frac{\cos(a)}{\sin(a)}

Learn it:Theorem 42.2.1 · More Trigonometric Identities

Book p. 114Checked numerically

5.3 Important Trigonometric Values

The book's table of values (unnumbered). The values from $90^\circ$ to $180^\circ$ follow from those below $90^\circ$ by $\sin(180^\circ - a) = \sin a$ and $\cos(180^\circ - a) = -\cos a$ .

$a$	$\cos a$	$\sin a$
$0^\circ$	$1$	$0$
$15^\circ$	$\frac{\sqrt3+1}{2\sqrt2}$	$\frac{\sqrt3-1}{2\sqrt2}$
$30^\circ$	$\frac{\sqrt3}{2}$	$\frac12$
$45^\circ$	$\frac{\sqrt2}{2}$	$\frac{\sqrt2}{2}$
$60^\circ$	$\frac12$	$\frac{\sqrt3}{2}$
$75^\circ$	$\frac{\sqrt3-1}{2\sqrt2}$	$\frac{\sqrt3+1}{2\sqrt2}$
$90^\circ$	$0$	$1$
$120^\circ$	$-\frac12$	$\frac{\sqrt3}{2}$
$135^\circ$	$-\frac{\sqrt2}{2}$	$\frac{\sqrt2}{2}$
$150^\circ$	$-\frac{\sqrt3}{2}$	$\frac12$
$180^\circ$	$-1$	$0$

Correction. The book prints $\cos 180^\circ = 1$ ; it is $-1$ (the point $(-1, 0)$ of the unit circle). It also prints the row $\cos 0^\circ = 1$ , $\sin 0^\circ = 0$ twice and labels the last sine $\sin 0^\circ$ instead of $\sin 180^\circ$ ; this table shows each angle once.

5.4 Unit Circle Identities

Theorem 5.4.1 · Unit Circle Identities

\sin(-a) = -\sin(a)

\sin(a) = \sin(180^\circ - a)

\cos(a) = \cos(-a)

\cos(a) = -\cos(180^\circ - a)

\tan(a) = -\tan(180^\circ - a)

\tan(-a) = -\tan(a)

Learn it:Theorem 42.4.1 · Unit Circle Identities Section 42.3 Special Angles and the Unit Circle

Book p. 116Checked numerically

5.5 Area of a Triangle using trigonometry

Theorem 5.5.1 · Area of a Triangle using trigonometry

In a triangle with side lengths $a$ , $b$ , $c$ , where the angle between sides $a$ and $b$ is denoted by $C$ ,

\text{Area of the triangle} = \frac{1}{2} \cdot ab \cdot \sin(C)

Learn it:Theorem 43.1.1 · Area of a Triangle Using Trigonometry

Book p. 117Checked numerically

5.6 Law of Sines

Theorem 5.6.1 · Law of Sines

In a triangle with sides $a$ , $b$ , $c$ and angles $A$ , $B$ , $C$ , where side $a$ is opposite angle $A$ , side $b$ is opposite angle $B$ , and side $c$ is opposite angle $C$ ,

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

where $R$ is the circumradius of the triangle.

Learn it:Theorem 43.2.1 · Law of Sines

Book p. 117Checked numerically

5.7 Law of Cosines

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Theorem 5.7.1 · Law of Cosines

In a triangle with side lengths $a$ , $b$ , $c$ , where the angle between sides $a$ and $b$ is denoted by $C$ ,

c^2 = a^2 + b^2 - 2ab \cdot \cos C

Learn it:Theorem 43.3.1 · Law of Cosines

Book p. 118Checked numerically

5.8 Pythagorean Identities

Theorem 5.8.1 · Pythagorean Identities

\sin^2(a) + \cos^2(a) = 1

\tan^2(a) + 1 = \sec^2(a)

\cot^2(a) + 1 = \csc^2(a)

Angle a40°

a = 40^\circ

P=(\cos a,\ \sin a)

f	f(40°)
cos a	$0.766$
sin a	$0.6428$
tan a	$0.8391$

Learn it:Theorem 42.5.1 · Pythagorean Identities

Book p. 119Checked numerically

5.9 Double Angle Identities

Theorem 5.9.1 · Double Angle Identities

\sin(2a) = 2\sin(a)\cos(a)

\cos(2a) = \cos^2(a) - \sin^2(a) = 2\cos^2(a) - 1 = 1 - 2\sin^2(a)

\tan(2a) = \frac{2\tan(a)}{1 - \tan^2(a)}

Learn it:Theorem 42.6.1 · Double Angle Identities

Book p. 119Checked numerically

5.10 Addition and Subtraction Identities

Theorem 5.10.1 · Addition and Subtraction Identities

\sin(a + b) = \sin(a)\cos(b) + \sin(b)\cos(a)

\sin(a - b) = \sin(a)\cos(b) - \sin(b)\cos(a)

\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)

\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)

\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a)\tan(b)}

\tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)}

Learn it:Theorem 42.7.1 · Addition and Subtraction Identities

Book p. 120Checked numerically

5.11 Half Angle Identities

Theorem 5.11.1 · Half Angle Identities

\sin\left(\frac a2\right) = \pm\sqrt{\frac{1 - \cos(a)}{2}}

\cos\left(\frac a2\right) = \pm\sqrt{\frac{1 + \cos(a)}{2}}

The sign is the sign of $\sin\frac a2$ (or $\cos\frac a2$ ): look at the quadrant of $\frac a2$ .

Learn it:Theorem 42.8.1 · Half Angle Identities

Book p. 121Checked numerically

5.12 Sum to Product Identities

Theorem 5.12.1 · Sum to Product Identities

\sin(a) + \sin(b) = 2\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)

\sin(a) - \sin(b) = 2\sin\left(\frac{a-b}{2}\right)\cos\left(\frac{a+b}{2}\right)

\cos(a) + \cos(b) = 2\cos\left(\frac{a-b}{2}\right)\cos\left(\frac{a+b}{2}\right)

\cos(a) - \cos(b) = -2\sin\left(\frac{a-b}{2}\right)\sin\left(\frac{a+b}{2}\right)

Learn it:Theorem 42.9.1 · Sum to Product Identities

Book p. 121Checked numerically

5.13 Product to Sum Identities

Theorem 5.13.1 · Product to Sum Identities

\sin(a)\sin(b) = \frac12\left(\cos(a - b) - \cos(a + b)\right)

\cos(a)\cos(b) = \frac12\left(\cos(a - b) + \cos(a + b)\right)

\sin(a)\cos(b) = \frac12\left(\sin(a + b) + \sin(a - b)\right)

Learn it:Theorem 42.10.1 · Product to Sum Identities

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Definition 5.13.2

A periodic function is a function that repeats a pattern of $y$ -values at regular intervals. One complete repetition of the pattern is called a cycle. The period of a function is the horizontal length of one complete cycle.

The period of $\sin$ and $\cos$ is $2\pi$ ; the period of $\tan$ is $\pi$ .

Correction

The book says the period of $\sin$ , $\cos$ and $\tan$ is $2\pi$ . Tangent repeats after only $\pi$ : $\tan(x + \pi) = \frac{-\sin x}{-\cos x} = \tan x$ . (The book also calls every periodic function "trigonometric"; periodic is the general word.)

Learn it:Definition 42.11.1 · Periodic function

Book p. 122Checked numerically

5.14 Periods and Graphs of Trigonometric Functions

Concept 5.14.1 · Sine Graph

The graph of $y = \sin x$ is a wave between $-1$ and $1$ . It starts at $(0, 0)$ , peaks at $x = \frac\pi2$ , crosses zero at every multiple of $\pi$ , and repeats every $2\pi$ .

Learn it:Concept 42.12.1 · Sine Graph

Book p. 122

Concept 5.14.2 · Cosine Graph

The graph of $y = \cos x$ is the sine wave shifted left by $\frac\pi2$ : $\cos x = \sin\left(x + \frac\pi2\right)$ . It starts at its peak $(0, 1)$ , crosses zero at odd multiples of $\frac\pi2$ , and repeats every $2\pi$ .

Learn it:Concept 42.12.2 · Cosine Graph

Book p. 122

Concept 5.14.3 · Tan Graph

The graph of $y = \tan x$ crosses zero at every multiple of $\pi$ and shoots off to $\pm\infty$ at the vertical asymptotes $x = \frac\pi2 + k\pi$ , where $\cos x = 0$ . It repeats every $\pi$ .

Learn it:Concept 42.12.3 · Tan Graph

Book p. 122

Remark 5.14.4

Long trigonometric expressions can be evaluated by telescoping, by using identities in clever ways, or by complex number substitutions (see Chapter 7, Complex Numbers).

Learn it:Remark 42.12.4 Section 44.3 Complex Numbers in Trigonometry

Book p. 123