An angle is a familiar figure in the world around us.

An angle is the union of two rays with a common endpoint called the vertex. In trigonometry, we often think of an angle as a rotation. To do so, think of locating a ray along the positive x-axis with its endpoint at the origin. This ray is called the initial side of the angle. Though we leave that ray fixed, think of making a copy of it and rotating it. A rotation counterclockwise is a positive rotation, and a rotation clockwise is a negative rotation. The ray at the end of the rotation is called the terminal side of the angle. The angle formed is said to be in standard position.

The measure of an angle or a rotation may be given in degrees. The Babylonians developed the idea of dividing the circumference of a circle into 360 equal parts, or degrees. If we let the measure of one of these parts be $1°$, then one complete positive revolution or rotation has a measure of $360°$. One half of a revolution has a measure of $180°$, one fourth of a revolution has a measure of $90°$, and so on. We can also speak of an angle of measure $60°$,$135°330°$, or $420°$. The terminal sides of these angles lie in quadrants I, II, IV, and I, respectively. The negative rotations $-30°$, $-110°$, and $-225°$ represent angles with terminal sides in quadrants IV, III, and II, respectively.

If two or more angles have the same terminal side, the angles are said to be coterminal. To find angles coterminal with a given angle, we add or subtract multiples of $360°$. For example, $420°$, shown above, has the same terminal side as $60°$, since $420°=60°+360°$. Thus we say that angles of measure $60°$ and $420°$ are coterminal. The negative rotation that measures $-300°$ is also coterminal with $60°$ because $60°-360°=-300°$. The set of all angles coterminal with $60°$ can be expressed as $60°+n\cdot 360°$, where $n$ is an integer. Other examples of coterminal angles shown above are $90°$ and $-270°$, $-90°$ and $270°$,$135°$ and $-225°$, $-30°$ and $330°$, and $-110°$ and $610°$.

Angles can be classified by their measures, as seen in the following figures.

Recall that two acute angles are complementary if the sum of their measures is $90°$. For example, angles that measure $10°$ and $80°$ are complementary because $10°+80°=90°$. Two positive angles are supplementary if the sum of their measures is $180°$. For example, angles that measure $45°$ and $135°$ are supplementary because $45°+135°=180°$.

Many applied problems in trigonometry involve the use of angles that are not acute. Thus we need to extend the domains of the trigonometric functions defined in Section 6.1 to angles, or rotations, of any size. To do this, we first consider a right triangle with one vertex at the origin of a coordinate system and one vertex on the positive x-axis. (See the figure at left.) The other vertex is at $P$, a point on the circle whose center is at the origin and whose radius r is the length of the hypotenuse of the triangle. This triangle is a reference triangle for angle $\theta$, which is in standard position. Note that $y$ is the length of the side opposite $\theta$ and x is the length of the side adjacent to $\theta$.

Recalling the definitions in Section 6.1, we note that three of the trigonometric functions of angle $\theta$ are defined as follows:

Since x and $y$ are the coordinates of the point $P$ and the length of the radius is the length of the hypotenuse, we can also define these functions as follows:

We will use these definitions for functions of angles of any measure. The following figures show angles whose terminal sides lie in quadrants II, III, and IV.

A reference triangle can be drawn for angles in any quadrant, as shown on the preceding page. Note that the angle is in standard position; that is, it is always measured from the positive half of the x-axis. The point $P\left(x,y\right)$ is a point, other than the vertex, on the terminal side of the angle. Each of its two coordinates may be positive, negative, or zero, depending on the location of the terminal side. The length of the radius, which is also the length of the hypotenuse of the reference triangle, is always considered positive. (Note that $x^2+y^2=r^2$, or $r=\sqrt{x^2+y^2}$.) Regardless of the location of $P$, we have the following definitions. We now extend the domain of the six trigonometric functions to include all angles, or rotations.

Values of the trigonometric functions can be positive, negative, or zero, depending on where the terminal side of the angle lies. Since the length of the radius is always positive, the signs of the function values depend only on the coordinates of the point $P$ on the terminal side of the angle. In the first quadrant, all function values are positive because both coordinates are positive. In the second quadrant, first coordinates are negative and second coordinates are positive; thus only the sine values and the cosecant values are positive. Similarly, we can determine the signs of the function values in the third and the fourth quadrants. Because of the reciprocal relationships, we need learn only the signs for the sine, cosine, and tangent functions.

Any point other than the origin on the terminal side of an angle in standard position can be used to determine the trigonometric function values of that angle. The function values are the same regardless of which point is used. To illustrate this, let’s consider an angle $\theta$ in standard position whose terminal side lies on the line $y=-{\displaystyle \frac{1}{2}}x$. We can determine two second-quadrant solutions of the equation, find the length r for each point, and then compare the sine, cosine, and tangent function values using each point.

• If $x=-4$, then $y=-{\displaystyle \frac{1}{2}}\left(-4\right)=2.$

• If $x=-8$, then $y=-{\displaystyle \frac{1}{2}}\left(-8\right)=4.$

• For $\left(-4,2\right),r=\sqrt{{\left(-4\right)}^2+2^2}=\sqrt{20}=2\sqrt{5}$.

• For $\left(-8,4\right),r=\sqrt{{\left(-8\right)}^2+4^2}=\sqrt{80}=4\sqrt{5}$.

Using $\left(-4,2\right)$ and $r=2\sqrt{5}$, we find that

Using $\left(-8,4\right)$ and $r=4\sqrt{5}$, we find that

We see that the function values are the same using either point. Any point other than the origin on the terminal side of an angle can be used to determine the trigonometric function values.

When we know one of the function values of an angle, we can find the other five if we know the quadrant in which the terminal side lies. The procedure is to sketch a reference triangle in the appropriate quadrant, use the Pythagorean equation as needed to find the lengths of its sides, and then find the ratios of the sides.

Example 4

Given that $\text{tan}\theta =-{\displaystyle \frac{2}{3}}$ and $\theta$ is in the second quadrant, find the other function values.

Solution

We first sketch a second-quadrant angle. Since

$$\text{tan}\theta ={\displaystyle \frac{y}{x}}=-{\displaystyle \frac{2}{3}}={\displaystyle \frac{2}{-3}},\text{Expressing}-{\displaystyle \frac{2}{3}}\text{as}{\displaystyle \frac{2}{-3}}\text{since}\theta \text{is in quadrant II}$$

we make the legs’ lengths 3 and 2. We measure off the 3 units in the negative direction since $\theta$ is in quadrant II. The hypotenuse must then have length $\sqrt{3^2+2^2}$, or $\sqrt{13}$. Now we read off the appropriate ratios:

$$\begin{array}{cc}\text{sin}\theta ={\displaystyle \frac{2}{\sqrt{13}}},\text{or}{\displaystyle \frac{2\sqrt{13}}{13}},\hfill & \text{csc}\theta ={\displaystyle \frac{\sqrt{13}}{2}},\hfill \\ \text{cos}\theta =-{\displaystyle \frac{3}{\sqrt{13}}},\text{or}-{\displaystyle \frac{3\sqrt{13}}{13}},\hfill & \text{sec}\theta =-{\displaystyle \frac{\sqrt{13}}{3}},\hfill \\ \text{tan}\theta =-{\displaystyle \frac{2}{3}},\hfill & \text{cot}\theta =-{\displaystyle \frac{3}{2}}.\hfill \end{array}$$

Now Try Exercise 35.

An angle whose terminal side falls on one of the axes is a quadrantal angle. One of the coordinates of any point on that side is 0. The definitions of the trigonometric functions still apply, but in some cases, function values will not be defined because a denominator will be 0.

Example 5

Find the sine, cosine, and tangent values for $90°$, $180°$, $270°$, and $360°$.

Solution

We first make a drawing of each angle in standard position and label a point on the terminal side. Since the function values are the same for all points on the terminal side, we choose $\left(0,1\right)$, $\left(-1,0\right)$, $\left(0,-1\right)$, and $\left(1,0\right)$ for convenience. Note that $r=1$ for each choice.

Then by the definitions we get

$$\begin{array}{cccc}\text{sin}90°={\displaystyle \frac{1}{1}}=1,\hfill & \text{sin}180°={\displaystyle \frac{0}{1}}=0,\hfill & \text{sin}270°={\displaystyle \frac{-1}{1}}=-1,\hfill & \text{sin}360°={\displaystyle \frac{0}{1}}=0,\hfill \\ \text{cos}90°={\displaystyle \frac{0}{1}}=0,\hfill & \text{cos}180°={\displaystyle \frac{-1}{1}}=-1,\hfill & \text{cos}270°={\displaystyle \frac{0}{1}}=0,\hfill & \text{cos}360°={\displaystyle \frac{1}{1}}=1,\hfill \\ \text{tan}90°={\displaystyle \frac{1}{0}},\mathbf{\text{Not defined}}\hfill & \text{tan}180°={\displaystyle \frac{0}{-1}}=0,\hfill & \text{tan}270°={\displaystyle \frac{-1}{0}},\mathbf{\text{Not defined}}\hfill & \text{tan}360°={\displaystyle \frac{0}{1}}=0.\hfill \end{array}$$

In Example 5, all the values can be found using a calculator, but you will find that it is convenient to be able to compute them mentally. It is also helpful to note that coterminal angles have the same function values. For example, $0°$ and $360°$ are coterminal; thus, $\text{sin}0°=0$, $\text{cos}0°=1$, and $\text{tan}0°=0.$

We can also mentally determine trigonometric function values whenever the terminal side makes a $30°$, $45°$, or $60°$ angle with the x-axis. Consider, for example, an angle of $150°$. The terminal side makes a $30°$ angle with the x-axis, since $180°-150°=30°$.

As the figure shows, $\Delta ONP$ is congruent to $\Delta ON\prime P\prime ;$ therefore, the ratios of the sides of the two triangles are the same. Thus the trigonometric function values are the same except perhaps for the sign. We could determine the function values directly from $\Delta ONP$, but this is not necessary. If we remember that in quadrant II, the sine is positive and the cosine and the tangent are negative, we can simply use the function values of $30°$ that we already know and prefix the appropriate sign. Thus,

Triangle ONP is the reference triangle, and the acute angle $\angle NOP$ is called a reference angle.

When the terminal side of an angle falls on one of the axes or makes a $30°,45°$, or $60°$ angle with the x-axis, we can find exact function values without the use of a calculator. But this group is only a small subset of all angles. Using a calculator, we can approximate the trigonometric function values of any angle. In fact, we can approximate or find exact function values of all angles without using a reference angle.

In many applications, we have a trigonometric function value and want to find the measure of a corresponding angle. When only acute angles are considered, there is only one angle for each trigonometric function value. This is not the case when we extend the domain of the trigonometric functions to the set of all angles. For a given function value, there is an infinite number of angles that have that function value. There can be two such angles for each value in the range from $0°$ to $360°$. To determine a unique answer in the interval $\left(0°,360°\right)$, the quadrant in which the terminal side lies must be specified.

The calculator gives the reference angle as an output for each function value that is entered as an input. Knowing the reference angle and the quadrant in which the terminal side lies, we can find the specified angle.