Angles

An angle consists of two rays with a common vertex. One of the rays is the initial side, and the other the terminal side. An angle can be viewed as a rotation of the initial side onto the terminal side. If the rotation is counterclockwise, the angle is positive; if the rotation is clockwise, the angle is negative.

Notation: The angle in the figure can be referred to as angle \(A O B\), or simply as angle \(O\), or as angle \(\theta\).

Angle Measure

The radian measure of an angle (abbreviated rad) is the length of the arc that the angle subtends in a circle of radius 1 , as shown in the figure.

The degree measure of an angle is the number of degrees in the angle, where a degree is \(\frac{1}{360}\) of a complete circle.

To convert degrees to radians, multiply by \(\pi / 180\). To convert radians to degrees, multiply by \(180 / \pi\).

Angles in Standard Position

An angle is in standard position if it is drawn in the \(x y\)-plane with its vertex at the origin and its initial side on the positive \(x\)-axis.

Two angles in standard position are coterminal if their terminal sides coincide. The reference angle \(\bar{\theta}\) associated with an angle \(\theta\) is the acute angle formed by the terminal side of \(\theta\) and the \(x\)-axis.

Length of an Arc; Area of a Sector

Consider a circle of radius \(r\).

The length \(s\) of an arc that subtends a central angle of \(\theta\) radians is \(s=r \theta\).

The area \(A\) of a sector with central angle of \(\theta\) radians is \(A=\frac{1}{2} r^{2} \theta\).

Circular Motion

Suppose a point moves along a circle of radius \(r\) and the ray from the center of the circle to the point traverses \(\theta\) radians in time \(t\). Let \(s=r \theta\) be the distance the point travels in time \(t\).

The angular speed of the point is \(\omega=\theta / t\). The linear speed of the point is \(v=s / t\). Linear speed \(v\) and angular speed \(\omega\) are related by the formula \(v=r \omega\).

Trigonometric Ratios

For a right triangle with an acute angle \(\theta\) the trigonometric ratios are defined as follows.

\[ \begin{array}{lll} \sin \theta=\frac{\text { opp }}{\text { hyp }} & \cos \theta=\frac{\text { adj }}{\text { hyp }} & \tan \theta=\frac{\text { opp }}{\text { adj }} \\ \csc \theta=\frac{\text { hyp }}{\text { opp }} & \sec \theta=\frac{\text { hyp }}{\text { adj }} & \cot \theta=\frac{\text { adj }}{\text { opp }} \end{array} \]

Trigonometric Functions of Angles

Let \(\theta\) be an angle in standard position, and let \(P(x, y)\) be a point on the terminal side, as shown in the figures. Let \(r=\sqrt{x^{2}+y^{2}}\) be the distance from the origin to the point \(P(x, y)\).

For nonzero values of the denominator the trigonometric functions are defined as follows.

\[ \begin{array}{lll} \sin t=\frac{y}{r} & \cos t=\frac{x}{r} & \tan t=\frac{y}{x} \\ \csc t=\frac{r}{y} & \sec t=\frac{r}{x} & \cot t=\frac{x}{y} \end{array} \]

Special Values of the Trigonometric Functions of Angles

The following table gives the values of the trigonometric functions at some special angles.

\(\boldsymbol{\theta}\) \(\boldsymbol{\theta}\) \(\boldsymbol{\operatorname { s i n } \boldsymbol { \theta }}\) \(\boldsymbol{\operatorname { c o s } \theta}\) \(\boldsymbol{\operatorname { t a n }} \boldsymbol{\theta}\) \(\boldsymbol{\operatorname { c s c } \theta}\) \(\boldsymbol{\operatorname { s e c } \theta}\) \(\boldsymbol{\operatorname { c o t } \theta \boldsymbol { \theta }}\)
\(0^{\circ}\) 0 0 1 0 - 1 -
\(30^{\circ}\) \(\frac{\pi}{6}\) \(\frac{1}{2}\) \(\frac{\sqrt{3}}{2}\) \(\frac{\sqrt{3}}{3}\) 2 \(\frac{2 \sqrt{3}}{3}\) \(\sqrt{3}\)
\(45^{\circ}\) \(\frac{\pi}{4}\) \(\frac{\sqrt{2}}{2}\) \(\frac{\sqrt{2}}{2}\) 1 \(\sqrt{2}\) \(\sqrt{2}\) 1
\(60^{\circ}\) \(\frac{\pi}{3}\) \(\frac{\sqrt{3}}{2}\) \(\frac{1}{2}\) \(\sqrt{3}\) \(\frac{2 \sqrt{3}}{3}\) 2 \(\frac{\sqrt{3}}{3}\)
\(90^{\circ}\) \(\frac{\pi}{2}\) 1 0 - 1 - 0
\(180^{\circ}\) \(\pi\) 0 -1 0 - -1 -
\(270^{\circ}\) \(\frac{3 \pi}{2}\) -1 0 - -1 - 0

Basic Trigonometric Identities

An identity is an equation that is true for all values of the variable. The basic trigonometric identities follow.

Reciprocal Identities

\[ \csc \theta=\frac{1}{\sin \theta} \quad \sec \theta=\frac{1}{\cos \theta} \quad \cot \theta=\frac{1}{\tan \theta} \]

Pythagorean Identities:

\[ \begin{aligned} \sin ^{2} \theta+\cos ^{2} \theta & =1 \\ \tan ^{2} \theta+1 & =\sec ^{2} \theta \\ 1+\cot ^{2} \theta & =\csc ^{2} \theta \end{aligned} \]

Area of a Triangle

The area \(\mathscr{A}\) of a triangle with sides of lengths \(a\) and \(b\) and with included angle \(\theta\) is

\[ \mathscr{A}=\frac{1}{2} a b \sin \theta \]

Inverse Trigonometric Functions

Inverse functions of the trigonometric functions have the domain and range shown in the following table.

Function Domain Range
\(\sin ^{-1}\) \([-1,1]\) \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)
\(\cos ^{-1}\) \([-1,1]\) \([0, \pi]\)
\(\tan ^{-1}\) \((-\infty, \infty)\) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

The inverse trigonometric functions are defined as follows.

\[ \begin{aligned} \sin ^{-1} x=y & \Leftrightarrow \quad \sin y=x \\ \cos ^{-1} x=y & \Leftrightarrow \quad \cos y=x \\ \tan ^{-1} x=y & \Leftrightarrow \quad \tan y=x \end{aligned} \]

The Law of Sines and the Law of Cosines

We follow the convention of labeling the angles of a triangle as \(A, B, C\) and the lengths of the corresponding opposite sides as \(a, b, c\), as labeled in the figure.

For a triangle \(A B C\) we have the following laws. The Law of Sines states that

\[ \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c} \]

The Law of Cosines states that

\[ \begin{aligned} & a^{2}=b^{2}+c^{2}-2 b c \cos A \\ & b^{2}=a^{2}+c^{2}-2 a c \cos B \\ & c^{2}=a^{2}+b^{2}-2 a b \cos C \end{aligned} \]

Heron's Formula

Let \(A B C\) be a triangle with sides \(a, b\), and \(c\).

Heron's Formula states that the area \(\mathscr{A}\) of triangle \(A B C\) is

\[ \mathscr{A}=\sqrt{s(s-a)(s-b)(s-c)} \]

where \(s=\frac{1}{2}(a+b+c)\) is the semiperimeter of the triangle.