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1.7: Trigonometric Functions of Any Angle

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

  • Identify the reference angles for angles in the unit circle.
  • Identify the ordered pair on the unit circle for angles whose reference angle is \begin{align*}30^\circ\end{align*}, \begin{align*}45^\circ\end{align*}, and \begin{align*}60^\circ\end{align*}, or a quadrantal angle, including negative angles, and angles whose measure is greater than \begin{align*}360^\circ\end{align*}.
  • Use these ordered pairs to determine values of trig functions of these angles.
  • Use calculators to find values of trig functions of any angle.

Reference Angles and Angles in the Unit Circle

In the previous lesson, one of the review questions asked you to consider the angle \begin{align*}150^\circ\end{align*}. If we graph this angle in standard position, we see that the terminal side of this angle is a reflection of the terminal side of \begin{align*}30^\circ\end{align*}, across the \begin{align*}y-\end{align*}axis.

Notice that \begin{align*}150^\circ\end{align*} makes a \begin{align*}30^\circ\end{align*} angle with the negative \begin{align*}x-\end{align*}axis. Therefore we say that \begin{align*}30^\circ\end{align*} is the reference angle for \begin{align*}150^\circ\end{align*}. Formally, the reference angle of an angle in standard position is the angle formed with the closest portion of the \begin{align*}x-\end{align*}axis. Notice that \begin{align*}30^\circ\end{align*} is the reference angle for many angles. For example, it is the reference angle for \begin{align*}210^\circ\end{align*} and for \begin{align*}-30^\circ\end{align*}.

In general, identifying the reference angle for an angle will help you determine the values of the trig functions of the angle.

Example 1: Graph each angle and identify its reference angle.

a. \begin{align*}140^\circ\end{align*}

b. \begin{align*}240^\circ\end{align*}

c. \begin{align*}380^\circ\end{align*}

Solution:

a. \begin{align*}140^\circ\end{align*} makes a \begin{align*}40^\circ\end{align*} angle with the \begin{align*}x-\end{align*}axis. Therefore the reference angle is \begin{align*}40^\circ\end{align*}.

b. \begin{align*}240^\circ\end{align*} makes a \begin{align*}60^\circ\end{align*} with the \begin{align*}x-\end{align*}axis. Therefore the reference angle is \begin{align*}60^\circ\end{align*}.

c. \begin{align*}380^\circ\end{align*} is a full rotation of \begin{align*}360^\circ\end{align*}, plus an additional \begin{align*}20^\circ\end{align*}. So this angle is co-terminal with \begin{align*}20^\circ\end{align*}, and \begin{align*}20^\circ\end{align*} is its reference angle.

If an angle has a reference angle of \begin{align*}30^\circ\end{align*}, \begin{align*}45^\circ\end{align*}, or \begin{align*}60^\circ\end{align*}, we can identify its ordered pair on the unit circle, and so we can find the values of the six trig functions of that angle. For example, above we stated that \begin{align*}150^\circ\end{align*} has a reference angle of \begin{align*}30^\circ\end{align*}. Because of its relationship to \begin{align*}30^\circ\end{align*}, the ordered pair for is \begin{align*}150^\circ\end{align*} is \begin{align*}\left ( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right )\end{align*}. Now we can find the values of the six trig functions of \begin{align*}150^\circ\end{align*}:

\begin{align*}\cos 150 & = x = \frac{-\sqrt{3}}{2} && \sec 150 = \frac{1}{x} = \frac{1}{\frac{-\sqrt{3}}{2}} = \frac{-2}{\sqrt{3}}\\ \sin 150 & = y = \frac{1}{2} && \csc 150 = \frac{1}{y} = \frac{1}{\frac{1}{2}} = 2\\ \tan 150 & = \frac{y}{x} = \frac{\frac{1}{2}}{\frac{-\sqrt{3}}{2}} = \frac{1}{-\sqrt{3}} && \cot 150 = \frac{x}{y} = \frac{\frac{-\sqrt{3}}{2}}{\frac{1}{2}} = -\sqrt{3}\end{align*}

Example 2: Find the ordered pair for \begin{align*}240^\circ\end{align*} and use it to find the value of sin \begin{align*}240^\circ\end{align*}.

Solution: \begin{align*}\sin 240^\circ = \frac{-\sqrt{3}}{2}\end{align*}

As we found in example 1, the reference angle for \begin{align*}240^\circ\end{align*} is \begin{align*}60^\circ\end{align*}. The figure below shows \begin{align*}60^\circ\end{align*} and the three other angles in the unit circle that have \begin{align*}60^\circ\end{align*} as a reference angle.

The terminal side of the angle \begin{align*}240^\circ\end{align*} represents a reflection of the terminal side of \begin{align*}60^\circ\end{align*} over both axes. So the coordinates of the point are \begin{align*}\left( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \right )\end{align*}. The \begin{align*}y-\end{align*}coordinate is the sine value, so \begin{align*}\sin 240^\circ = \frac{-\sqrt{3}}{2}\end{align*}.

Just as the figure above shows \begin{align*}60^\circ\end{align*} and three related angles, we can make similar graphs for \begin{align*}30^\circ\end{align*} and \begin{align*}45^\circ\end{align*}.

Knowing these ordered pairs will help you find the value of any of the trig functions for these angles.

Example 3: Find the value of \begin{align*}\cot 300^\circ\end{align*}

Solution: \begin{align*}\cot 300^\circ -\frac{1}{\sqrt{3}}\end{align*}

Using the graph above, you will find that the ordered pair is \begin{align*}\left ( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right )\end{align*}. Therefore the cotangent value is \begin{align*}\cot 300 = \frac{x}{y} = \frac{\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{2} \times -\frac{2}{\sqrt{3}} = -\frac{1}{\sqrt{3}}\end{align*}

We can also use the concept of a reference angle and the ordered pairs we have identified to determine the values of the trig functions for other angles.

Trigonometric Functions of Negative Angles

Recall that graphing a negative angle means rotating clockwise. The graph below shows \begin{align*}-30^\circ\end{align*}.

Notice that this angle is coterminal with \begin{align*}330^\circ\end{align*}. So the ordered pair is \begin{align*}\left ( \frac{\sqrt{3}}{2}, -\frac{1}{2} \right )\end{align*}. We can use this ordered pair to find the values of any of the trig functions of \begin{align*}-30^\circ\end{align*}. For example, \begin{align*}\cos (-30^\circ) = x = \frac{\sqrt{3}}{2}\end{align*}.

In general, if a negative angle has a reference angle of \begin{align*}30^\circ\end{align*}, \begin{align*}45^\circ\end{align*}, or \begin{align*}60^\circ\end{align*}, or if it is a quadrantal angle, we can find its ordered pair, and so we can determine the values of any of the trig functions of the angle.

Example 4: Find the value of each expression.

a. \begin{align*}\sin(-45^\circ)\end{align*}

b. \begin{align*}\sec(-300^\circ)\end{align*}

c. \begin{align*}\cos(-90^\circ)\end{align*}

Solution:

a. \begin{align*}\sin (-45^\circ) = -\frac{\sqrt{2}}{2}\end{align*}

\begin{align*}-45^\circ\end{align*} is in the \begin{align*}4^{th}\end{align*} quadrant, and has a reference angle of \begin{align*}45^\circ\end{align*}. That is, this angle is coterminal with \begin{align*}315^\circ\end{align*}. Therefore the ordered pair is \begin{align*}\left ( \frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} \right )\end{align*} and the sine value is \begin{align*}-\frac{\sqrt{2}}{2}\end{align*}.

b. \begin{align*}\sec(-300^\circ) = 2\end{align*}

The angle \begin{align*}-300^\circ\end{align*} is in the \begin{align*}1^{st}\end{align*} quadrant and has a reference angle of \begin{align*}60^\circ\end{align*}. That is, this angle is coterminal with \begin{align*}60^\circ\end{align*}. Therefore the ordered pair is \begin{align*}\left ( \frac{1}{2}, \frac{\sqrt{3}}{2} \right )\end{align*} and the secant value is \begin{align*}\frac{1}{x} = \frac{1}{\frac{1}{2}} = 2\end{align*}.

c. \begin{align*}\cos(-90^\circ) = 0\end{align*}

The angle \begin{align*}-90^\circ\end{align*} is coterminal with \begin{align*}270^\circ\end{align*}. Therefore the ordered pair is (0, -1) and the cosine value is 0.

We can also use our knowledge of reference angles and ordered pairs to find the values of trig functions of angles with measure greater than 360 degrees.

Trigonometric Functions of Angles Greater than 360 Degrees

Consider the angle \begin{align*}390^\circ\end{align*}. As you learned previously, you can think of this angle as a full 360 degree rotation, plus an additional 30 degrees. Therefore \begin{align*}390^\circ\end{align*} is coterminal with \begin{align*}30^\circ\end{align*}. As you saw above with negative angles, this means that \begin{align*}390^\circ\end{align*} has the same ordered pair as \begin{align*}30^\circ\end{align*}, and so it has the same trig values. For example,

\begin{align*}\cos 390^\circ = \cos 30^\circ = \frac{\sqrt{3}}{2}\end{align*}

In general, if an angle whose measure is greater than \begin{align*}360\end{align*} has a reference angle of \begin{align*}30^\circ\end{align*}, \begin{align*}45^\circ\end{align*}, or \begin{align*}60^\circ\end{align*}, or if it is a quadrantal angle, we can find its ordered pair, and so we can find the values of any of the trig functions of the angle. Again, determine the reference angle first.

Example 5: Find the value of each expression.

a. \begin{align*}\sin 420^\circ\end{align*}

b. \begin{align*}\tan 840^\circ\end{align*}

c. \begin{align*}\cos 540^\circ\end{align*}

Solution:

a. \begin{align*}\sin 420^\circ = \frac{\sqrt{3}}{2}\end{align*}

\begin{align*}420^\circ\end{align*} is a full rotation of 360 degrees, plus an additional 60 degrees. Therefore the angle is coterminal with \begin{align*}60^\circ\end{align*}, and so it shares the same ordered pair, \begin{align*}\left ( \frac{1}{2}, \frac{\sqrt{3}}{2} \right )\end{align*}. The sine value is the \begin{align*}y-\end{align*}coordinate.

b. \begin{align*}\tan 840^\circ = -\sqrt{3}\end{align*}

\begin{align*}840^\circ\end{align*} is two full rotations, or 720 degrees, plus an additional 120 degrees:

\begin{align*}840 = 360 + 360 + 120\end{align*}

Therefore \begin{align*}840^\circ\end{align*} is coterminal with \begin{align*}120^\circ\end{align*}, so the ordered pair is \begin{align*}\left ( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right )\end{align*}. The tangent value can be found by the following:

\begin{align*}\tan 840^\circ = \tan 120^\circ = \frac{y}{x} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \frac{\sqrt{3}}{2} \times -\frac{2}{1} = -\sqrt{3}\end{align*}

c. \begin{align*}\cos 540^\circ = -1\end{align*}

\begin{align*}540^\circ\end{align*} is a full rotation of 360 degrees, plus an additional 180 degrees. Therefore the angle is coterminal with \begin{align*}180^\circ\end{align*}, and the ordered pair is (-1, 0). So the cosine value is -1.

So far all of the angles we have worked with are multiples of 30, 45, 60, and 90. Next we will find approximate values of the trig functions of other angles.

Using a Calculator to Find Values

If you have a scientific calculator, you can determine the value of any trig function for any angle. Here we will focus on using a TI graphing calculator to find values.

First, your calculator needs to be in the correct “mode.” In chapter 2 you will learn about a different system for measuring angles, known as radian measure. In this chapter, we are measuring angles in degrees. We need to make sure that the calculator is in degrees. To do this, press \begin{align*}\fbox{\text{MODE}}\end{align*}. In the third row, make sure that Degree is highlighted. If Radian is highlighted, scroll down to this row, scroll over to Degree, and press \begin{align*}\fbox{\text{ENTER}}\end{align*}. This will highlight Degree. Then press \begin{align*} \begin{array} {|c|} \hline 2^{\text{nd}} \\ \hline \end{array}\end{align*} \begin{align*}\fbox{\text{MODE}}\end{align*} to return to the main screen.

Now you can calculate any value. For example, we can verify the values from the table above. To find \begin{align*}\sin 130^\circ\end{align*}, press \begin{align*}\fbox{\text{Sin}}\end{align*} \begin{align*}\fbox{130}\end{align*} \begin{align*}\fbox{\text{ENTER}}\end{align*}. The calculator should return the value .7660444431.

Example 6: Find the approximate value of each expression. Round your answer to 4 decimal places.

a. \begin{align*}\sin 130^\circ\end{align*}

b. \begin{align*}\cos 15^\circ\end{align*}

c. \begin{align*}\tan 50^\circ\end{align*}

Solution:

a. \begin{align*}\sin 130^\circ \approx 0.7660\end{align*}

b. \begin{align*}\cos 15^\circ \approx 0.9659\end{align*}

c. \begin{align*}\tan 50^\circ \approx 1.1918\end{align*}

You may have noticed that the calculator provides a “(“ after the SIN. In the previous calculations, you can actually leave off the “)”. However, in more complicated calculations, leaving off the closing “)” can create problems. It is a good idea to get in the habit of closing parentheses.

You can also use a calculator to find values of more complicated expressions.

Example 7: Use a calculator to find an approximate value of \begin{align*}\sin 25^\circ + \cos 25^\circ\end{align*}. Round your answer to 4 decimal places.

Solution: \begin{align*}\sin 25^\circ + \cos 25^\circ \approx 1.3289\end{align*}

\begin{align*}^* \text{This is an example where you need to close the parentheses.}\end{align*}

Points to Consider

  • What is the difference between the measure of an angle, and its reference angle? In what cases are these measures the same value?
  • Which angles have the same cosine value, or the same sine value? Which angles have opposite cosine and sine values?

Review Questions

  1. State the reference angle for each angle.
    1. \begin{align*}190^\circ\end{align*}
    2. \begin{align*}-60^\circ\end{align*}
    3. \begin{align*}1470^\circ\end{align*}
    4. \begin{align*}-135^\circ\end{align*}
  2. State the ordered pair for each angle.
    1. \begin{align*}300^\circ\end{align*}
    2. \begin{align*}-150^\circ\end{align*}
    3. \begin{align*}405^\circ\end{align*}
  3. Find the value of each expression.
    1. \begin{align*}\sin 210^\circ\end{align*}
    2. \begin{align*}\tan 270^\circ\end{align*}
    3. \begin{align*}\csc 120^\circ\end{align*}
  4. Find the value of each expression.
    1. \begin{align*}\sin 510^\circ\end{align*}
    2. \begin{align*}\cos 930^\circ\end{align*}
    3. \begin{align*}\csc 405^\circ\end{align*}
  5. Find the value of each expression.
    1. \begin{align*}\cos -150^\circ\end{align*}
    2. \begin{align*}\tan -45^\circ\end{align*}
    3. \begin{align*}\sin -240^\circ\end{align*}
  6. Use a calculator to find each value. Round to 4 decimal places.
    1. \begin{align*}\sin 118^\circ\end{align*}
    2. \begin{align*}\tan 55^\circ\end{align*}
    3. \begin{align*}\cos 100^\circ\end{align*}
  7. Recall, in lesson 1.4, we introduced inverse trig functions. Use your calculator to find the measure of an angle whose sine value is 0.2.
  8. In example 6c, we found that \begin{align*}\tan 50^\circ \approx 1.1918\end{align*}. Use your knowledge of a special angle to explain why this value is reasonable. HINT: You will need to change the tangent of this angle into a decimal.
  9. Use the table below or a calculator to explore sum and product relationships among trig functions. Consider the following functions: \begin{align*}f(x) & = \sin (x+x) \ \text{and} \ g(x) = \sin (x) + \sin (x)\\ h(x) & = \sin (x) \ ^* \ \sin(x) \ \text{and} \ j(x) = \sin (x^2)\end{align*} Do you observe any patterns in these functions? Are there any equalities among the functions? Can you make a general conjecture about \begin{align*}\sin (a) + \sin (b)\end{align*} and \begin{align*}\sin (a+b)\end{align*} for all values of \begin{align*}a, b\end{align*}? What about \begin{align*}\sin(a) \sin(a)\end{align*} and \begin{align*}\sin (a^2)\end{align*}?
\begin{align*}a^\circ\end{align*} \begin{align*}b^\circ\end{align*} \begin{align*}\sin a + \sin b\end{align*} \begin{align*}\sin(a + b)\end{align*}
10 30 .6736 .6428
20 60 1.2080 .9848
55 78 1.7973 .7314
122 25 1.2707 .5446
200 75 .6239 -.9962
  1. Use a calculator or your knowledge of special angles to fill in the values in the table, then use the values to make a conjecture about the relationship between \begin{align*}(\sin a)^2\end{align*} and \begin{align*}(\cos a)^2\end{align*}. If you use a calculator, round all values to 4 decimal places.
\begin{align*}a\end{align*} \begin{align*}(\sin a)^2\end{align*} \begin{align*}(\cos a)^2\end{align*}
0
25
45
80
90
120
250

Review Answers

    1. \begin{align*}10^\circ\end{align*}
    2. \begin{align*}60^\circ\end{align*}
    3. \begin{align*}30^\circ\end{align*}
    4. \begin{align*}45^\circ\end{align*}
    1. \begin{align*}\left ( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right )\end{align*}
    2. \begin{align*}\left ( -\frac{\sqrt{3}}{2}, -\frac{1}{2} \right )\end{align*}
    3. \begin{align*}\left ( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right )\end{align*}
    1. \begin{align*}-\frac{1}{2}\end{align*}
    2. 0
    3. \begin{align*}\frac{2}{\sqrt{3}}\end{align*}
    1. \begin{align*}\frac{1}{2}\end{align*}
    2. \begin{align*}-\frac{\sqrt{3}}{2}\end{align*}
    3. \begin{align*}\sqrt{2}\end{align*}
    1. \begin{align*}-\frac{\sqrt{3}}{2}\end{align*}
    2. -1
    3. \begin{align*}\frac{\sqrt{3}}{2}\end{align*}
    1. 0.8828
    2. 1.4281
    3. -0.1736
  1. About 11.54 degrees or about 168.46 degrees.
  2. This is reasonable because \begin{align*}\tan 45^\circ = 1\end{align*} and the \begin{align*}\tan 60^\circ = \sqrt{3} \approx 1.732\end{align*}, and the \begin{align*}\tan 50^\circ\end{align*} should fall between these two values.
  3. Conjecture: \begin{align*}\sin a + \sin b \ne \sin(a + b)\end{align*}
\begin{align*}a\end{align*} \begin{align*}(\sin a)^2\end{align*} \begin{align*}(\cos a)^2\end{align*}
0 0 1
25 .1786 .8216
45 \begin{align*}\frac{1}{2}\end{align*} \begin{align*}\frac{1}{2}\end{align*}
80 .9698 .0302
90 1 0
120 .75 .25
235 .6710 .3290
310 .5898 .4132

Conjecture: \begin{align*}(\sin a)^2 + (\cos a)^2 = 1\end{align*}.

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