We have discussed finding the sine and cosine for angles in the first quadrant, but what if our angle is in another quadrant? For any given angle in the first quadrant, there is an angle in the second quadrant with the same sine value. Because the sine value is the y -coordinate on the unit circle, the other angle with the same sine will share the same y -value, but have the opposite x -value. Likewise, there will be an angle in the fourth quadrant with the same cosine as the original angle.
The angle with the same cosine will share the same x -value but will have the opposite y -value. Suppose a rider snaps a photograph while stopped twenty feet above ground level. The rider then rotates three-quarters of the way around the circle. To answer questions such as this one, we need to evaluate the sine or cosine functions at angles that are greater than 90 degrees or at a negative angle.
Reference angles make it possible to evaluate trigonometric functions for angles outside the first quadrant. We will use the reference angle of the angle of rotation combined with the quadrant in which the terminal side of the angle lies.
We can find the cosine and sine of any angle in any quadrant if we know the cosine or sine of its reference angle. The absolute values of the cosine and sine of an angle are the same as those of the reference angle. The sign depends on the quadrant of the original angle. The cosine will be positive or negative depending on the sign of the x -values in that quadrant. The sine will be positive or negative depending on the sign of the y -values in that quadrant. Angles have cosines and sines with the same absolute value as their reference angles.
The sign positive or negative can be determined from the quadrant of the angle. The y -coordinate is positive, so the sine value is positive. Now that we have learned how to find the cosine and sine values for special angles in the first quadrant, we can use symmetry and reference angles to fill in cosine and sine values for the rest of the special angles on the unit circle. They are shown in Figure First we find the reference angle corresponding to the given angle.
Then we take the sine and cosine values of the reference angle , and give them the signs corresponding to the y — and x -values of the quadrant. We must determine the appropriate signs for x and y in the given quadrant. Skip to main content. Module 7: Trigonometric Functions. Search for:. Identify the domain and range of sine and cosine functions. Use reference angles to evaluate trigonometric functions. Evaluate sine and cosine values using a calculator.
A degree shift in a sinusoidal function is indistinguishable from a phase reversal. If you don't use it, you are using radians. Add a comment. Active Oldest Votes. Nice to see you here. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password.
Post as a guest Name. Email Required, but never shown. Upcoming Events. Then we take the sine and cosine values of the reference angle , and give them the signs corresponding to the y — and x -values of the quadrant. We must determine the appropriate signs for x and y in the given quadrant. Explain how the cosine of an angle in the second quadrant differs from the cosine of its reference angle in the unit circle. Explain how the sine of an angle in the second quadrant differs from the sine of its reference angle in the unit circle.
For the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places. For the following exercises, use a graphing calculator to evaluate.
For the following exercises, use this scenario: A child enters a carousel that takes one minute to revolve once around. Assume the carousel revolves counter clockwise. There are multiple answers. Skip to main content. Module 1: Trigonometric Functions. Search for:. Unit Circle: Sine and Cosine Functions To define our trigonometric functions, we begin by drawing a unit circle, a circle centered at the origin with radius 1, as shown in Figure 2.
Figure 3. Figure 4. Figure 5. Figure 6. Figure 7. Figure 8. Figure 9. Figure How To: Given an angle in radians, use a graphing calculator to find the cosine. If the calculator has degree mode and radian mode, set it to radian mode. Press the COS key. Analysis of the Solution We can find the cosine or sine of an angle in degrees directly on a calculator with degree mode.
An angle in the first quadrant is its own reference angle. How To: Given an angle in standard position, find the reference angle, and the cosine and sine of the original angle. Measure the angle between the terminal side of the given angle and the horizontal axis. That is the reference angle. Determine the values of the cosine and sine of the reference angle. Give the cosine the same sign as the x -values in the quadrant of the original angle.
Give the sine the same sign as the y -values in the quadrant of the original angle.
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