Evaluating the six trigonometric functions

In the Trigonometric Functions section, you will learn how to evaluate trigonometric functions at various angle measures and also graph trigonometric functions.

Has no one condemned you? Summary: In this section, you will: Evaluate trigonometric functions of any angle. Find reference angles. The London Eye is a Ferris wheel with a diameter of feet. By combining the ideas of the unit circle and right triangles, the location of any capsule on the Eye can be described with trigonometry. Lesson looked at the unit circle. Lesson explored right triangle trigonometry.

Evaluating the six trigonometric functions

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All the unit circle formulas can be similarly modified.

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Evaluating the six trigonometric functions

Trigonometric functions are used to model many phenomena, including sound waves, vibrations of strings, alternating electrical current, and the motion of pendulums. In fact, almost any repetitive, or cyclical, motion can be modeled by some combination of trigonometric functions. In this section, we define the six basic trigonometric functions and look at some of the main identities involving these functions. To use trigonometric functions, we first must understand how to measure the angles. Although we can use both radians and degrees, radians are a more natural measurement because they are related directly to the unit circle, a circle with radius 1. The radian measure of an angle is defined as follows. We say the angle corresponding to the arc of length 1 has radian measure 1. Trigonometric functions allow us to use angle measures, in radians or degrees, to find the coordinates of a point on any circle—not only on a unit circle—or to find an angle given a point on a circle. They also define the relationship between the sides and angles of a triangle. The trigonometric functions are then defined as.

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Lesson looked at the unit circle. Summary: In this section, you will: Evaluate trigonometric functions of any angle. Has no one condemned you? Figure 3 Solution Find r. In the Trigonometric Functions section, you will learn how to evaluate trigonometric functions at various angle measures and also graph trigonometric functions. Find the reference angle using the appropriate reference angle formula from the first portion of this review section. You will learn in Section 1. Since the x is negative and r are both positive, cosine is negative. The graph of the function is symmetrical about the y -axis. Start by choosing a point on the terminal sides of the angle. When the original angle is given in quadrant two, three, or four, a reference angle should be found. Figure 5 shows which trigonometric functions are positive in each quadrant.

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Since the x is negative and r are both positive, cosine is negative. All the ideas from this lesson can be combined to evaluate trigonometric functions of any real number. Figure 4: Points for quadrantal angles. The graph is not symmetrical about the y -axis. The quadrant of the original angle determines whether the answer is positive or negative. All the unit circle formulas can be similarly modified. You will learn in Section 1. All along the graph, any two points with opposite x -values also have opposite y -values. Comparing the unit circle formulas and the right triangle formulas develops the formulas for any angle. Find reference angles. Since the y and r are both positive, sine is positive.