Nice Angles Study Guide
We now evaluate the trigonometric values of angles that are larger than 90° and less than 0°. The sine and cosine come directly from the y- and x-coordinates of the point on the unit circle, which correspond to the given angle.
The nice angles are the multiples of 30°, 45°, 60°, and 90° . We can find the point on the unit circle, which corresponds to any nice angle. The coordinates of this point are trigonometric values of the angle.
For example, suppose θ is an angle between 0° and 90°. This angle must correspond to some point (x,y) on the unit circle. This means that there is a right triangle with hypotenuse 1, opposite side y, and adjacent side x, as shown in Figure 11.1.
The sine of this angle is . The cosine of this angle is . In other words, the x-coordinate is the cosine and the y-coordinate is the sine, as shown in Figure 11.2. We use this to define the sine and cosine of every angle, even negative angles and angles greater than 90°.
An angle of 120° is a little more (30°) than 90°. Thus, it looks like the angle in Figure 11.3. The y-coordinate is longer, so and . Because the point is above and to the left of the origin, the point is . The x-coordinate is the cosine of the angle, so cos(120°) = .
The point of the unit circle that corresponds to 90° is (0,1), as shown in Figure 11.4.
Because the sine is the same as the y-coordinate of the point, sin(90°) = 1.
Find the sine and cosine of .
is the same as .
This angle corresponds with the point , as shown in Figure 11.5.
The y-coordinate is the sine and the x-coordinate is the cosine, so
Other Trigonometric Functions of Nice Angles
Once we know the sine and cosine of an angle θ, we can put them together to evaluate any other trigonometric function. We use the following formulas:
What is sec(l35°)?
The angle l35° corresponds to the point , as shown in Figure 11.6.
is the same as
which corresponds to , as shown in Figure 11.7.
This means and , so
Find tan(90°). The point (0,1) on the unit circle corresponds to the 90° angle. Thus cos(90°) = 0 and sin(90°) = 1. The tangent of 90° should be
except that division by zero is never allowed.
The only way out of this mess is to declare that tan(90°) is undefined. Similarly, sec(90°) is also not defined because it would involve division by cos(90°) = 0 if it did exist. At each multiple of 90° (such as 90° , 180° ,270°, 0°, –90°, -180°, and so on), there will be undefined functions because one of the coordinates (sine or cosine) will be zero.
Graphing Sine, Cosine, and Tangent
Now that we know the sine of many angles, we can use this to sketch the graph of y = sin(x) shown in Figure 11.8.
This graph oscillates up and down forever between 1 and –1. As the angles on the x-axis increase, the graph gives the height of the corresponding point on the unit circle, rising up, then down, and then repeating.
The graph of y = cos(x), shown in Figure 11.9, is almost the same, but shifted over by 90°, or π, radians. This is because the side-to-side motion of a point going around the unit circle follows the same pattern as its up-and-down motion.
The functions sin(x) and cos(x) are said to be periodic with period 2π because they repeat every 2π = 360° (every full circle). The graph of y = tan(x), shown in Figure 11.10, is more curious. Because , it is undefined everywhere cos(x) = 0. When cos(x) gets close to zero, the values of tan(x) get really big and approach vertical asymptotes, at , and at every other place that could be described as where k is an integer.
The tangent function is also periodic, but the period is π
Practice problems for this study guide can be found at:
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