Exercise 11 Page 963

We want to find all the angles whose tangent is sqrt(3). Since we know the value of tangent but not the value of the angle, the process of finding the missing value uses an inverse trigonometric function. tan^(- 1) ( sqrt(3)) First, we need to use the definition of tangent to find the appropriate reference angle. The tangent of an angle is defined as the ratio of the sine to the cosine of the angle. tan θ =sin θ/cos θ Let's do it!

The tangent of an angle that measures 60^(∘) is sqrt(3). Let's now construct a 30^(∘)-60^(∘)-90^(∘) triangle. Since we will also need to use the unit circle, our right triangle will have a hypotenuse that measures 1.

In this type of special triangle, the length of the shorter leg is half the length of the hypotenuse, and the length of the longer leg is sqrt(3) times the length of the shorter leg.

Next, let's locate this triangle in the coordinate plane. Since the given value is positive, our solutions will be located in the quadrants that have positive tangent values — Quadrant I and Quadrant III.

We can draw a 30^(∘)-60^(∘)-90^(∘) triangle on a unit circle with the measurements found above in these quadrants. We will place the 60^(∘) angle at the origin.

Finally, we will use these triangles to locate two angles whose tangent is sqrt(3). Keep in mind that a half turn measures 180^(∘). Since the first angle is in the Quadrant I, it is already simplified. Therefore, we will add 60 to 180 to get the second angle. 180+ 60&= 240^(∘) Let's show the obtained angles on the coordinate plane.

We found that tan ( sqrt(3)) can be either 60^(∘) or 240^(∘). Finally, keep in mind that if we add or subtract a multiple of 360^(∘), the terminal side of the angle will be in the same position. Therefore, the tangent of the resulting angles will also be sqrt(3). 60^(∘)+360^(∘) * n and 240^(∘)+360^(∘) * n, wheren is any integer