Trigonometric Function Values in Tables

As you work through this chapter, you will learn about different applications of the trig functions. In many cases, you will need to find the value of a function of an angle that is not necessarily one of the "special" angles we have worked with so far. Traditionally, textbooks have provided students with tables that contain values of the trig functions. Below is a table that provides approximate values of the sine, cosine, and tangent values of several angles.

Angle (^\circ) Cosine Sine Tangent
0 1.0000 0.0000 0.0000
5 0.9962 0.0872 0.0875
10 0.9848 0.1736  0.1763
15 0.9659 0.2588 0.2679
20 0.9397 0.3420 0.3640
25 0.9063 0.4226 0.4663
30 0.8660  0.5000 0.5774
35 0.8192 0.5736 0.7002
40 0.7660 0.6428 0.8391
45 0.7071 0.7071 1.0000
50 0.6428 0.7660 1.1918
55 0.5736 0.8192 1.4281
60 0.5000 0.8660 1.7321
65 0.4226 0.9063 2.1445
70 0.3420 0.9397 2.7475
75 0.2588 0.9659 3.7321
80 0.1736 0.9848 5.6713
85 0.0872 0.9962 11.4301
90 0.0000 1.0000 undefined
95 -0.0872 0.9962 -11.4301
100  -0.1736 0.9848 -5.6713
105 -0.2588 0.9659 -3.7321
110  -0.3420 0.9397 -2.7475
115 -0.4226 0.9063 -2.1445
120 -0.5000 0.8660 -1.7321
125 -0.5736 0.8192 -1.4281
130 -0.6428 0.7660 -1.1918
135 -0.7071 0.7071 -1.0000
140 -0.7660 0.6428 -0.8391
145 -0.8192 0.5736 -0.7002
150 -0.8660 0.5000 -0.5774
155 -0.9063 0.4226 -0.4663
160  -0.9397 0.3420  -0.3640
165  -0.9659 0.2588 -0.2679
170 -0.9848 0.1736 -0.1763
175 -0.9962 0.0872 -0.0875
 180 -1.0000 0.0000 0.0000

We can use the table to identify approximate values.

Example 6: Find the approximate value of each expression, using the table above.

a. \sin(130^\circ )

b. \cos(15^\circ )

c. \tan(50^\circ )

Solution:

a. \sin(130^\circ )\approx 0.7660

We can identify the sine value by finding the row in the table for 130\;\mathrm{degrees}. The sine value is found in the third row of the table. Note that this is an approximate value. We can evaluate the reasonableness of this value by thinking about an angle that is close to 130\;\mathrm{degrees}, 120\;\mathrm{degrees}. We know that the ordered pair for 120 is \left (-\frac{1} {2},\frac{\sqrt{3}} {2}\right ), so the sine value is \frac{\sqrt{3}} {2} \approx 0.8660, which is also in the table. It is reasonable that \sin(130^\circ )\approx0.7660, which is slightly less than the sine value of 120, given where the terminal sides of these angles intersect the unit circle.

b. \cos(15^\circ ) \approx 0.9659

We can identify this cosine value by finding the row for 15\;\mathrm{degrees}. The cosine value is found in the second column. Again, we can determine if this value is reasonable by considering a nearby angle. 15^\circ is between 0^\circ and 30^\circ, and its cosine value is between the cosine values of these two angles.

c. \tan(50^\circ ) \approx 1.1918

We can identify this tangent value by finding the row for 50\;\mathrm{degrees}, and reading the final column of the table. In the review questions, you will be asked to explain why the tangent value seems reasonable.

Using a Calculator to Find Values

Video: Determining trig function values on the calculator

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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. (This is analogous to measuring distance in miles or in kilometers. It's just a different system of measurement.) We need to make sure that the calculator is working in degrees. To do this, press [MODE]. You will see that the third row says Radian Degree. If Degree is highlighted, you are in the correct mode. If Radian is highlighted, scroll down to this row, scroll over to Degree, and press [ENTER]. This will highlight Degree. Then press 2^{nd} [Mode] 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 \sin(130^\circ ), press [SIN][130][ENTER]. The calculator should return the value .7660444431.

You may have noticed that the calculator provides a "(" after the SIN. In the previous calculation, 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 \sin(25^\circ ) + \cos(25^\circ ). Round your answer to 4\;\mathrm{decimal} places.

Solution:\sin(25^\circ ) + \cos(25^\circ )\approx 1.3289

To use a TI graphing calculator, press [SIN][25][+][COS][25][ENTER]. The calculator should return the number 1.328926049. This rounds to 1.3289.

 

How does the calculator determine trig function values?

Related articles

http://www.homeschoolmath.net/teaching/sine_calculator.php http://en.wikipedia.org/wiki/Exact_trigonometric_constants