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. Understanding how to find a reference angle of a given angle is an important skill needed to evaluate trigonometric functions and is reviewed here.

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.

Evaluating the six trigonometric functions

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Summary: In this section, you will: Evaluate trigonometric functions of any angle. Thus, if you can evaluate sine and cosine at various angle values, you can also evaluate the other trigonometric functions at various angle values. Skip to main content.

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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.

Evaluating the six trigonometric functions

Everest, which straddles the border between China and Nepal, is the tallest mountain in the world. Measuring its height is no easy task and, in fact, the actual measurement has been a source of controversy for hundreds of years. The measurement process involves the use of triangles and a branch of mathematics known as trigonometry. In this section, we will define a new group of functions known as trigonometric functions, and find out how they can be used to measure heights, such as those of the tallest mountains. If we drop a line segment vertically down from this point to the x axis, we would form a right triangle inside of the circle. Triangles obtained from different radii will all be similar triangles, meaning corresponding sides scale proportionally. While the lengths of the sides may change, the ratios of the side lengths will always remain constant for any given angle. To be able to refer to these ratios more easily, we will give them names.

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CC licensed content, Original. Use the Pythagorean Theorem to find y. To be able to use our six trigonometric functions freely with both positive and negative angle inputs, we should examine how each function treats a negative input. Find reference angles. Evaluate the six trigonometric functions based on the given point on the terminal side of an angle in standard position. It is easiest to start by stretching a graph of the angle in standard position like in figure 7. Determine the appropriate sign of your found value for cosine or sine based on the quadrant of the original angle. The values of the trigonometric functions of the angle in standard position equal values of the trigonometric functions of the reference angle with the appropriate negative signs for the quadrant. Since the formula from the unit circle and the right triangles give the same expressions for example 1, the acute angle by the origin in the triangle and the angle in standard position must give the same values of the trigonometric functions. Has no one condemned you? We can evaluate trigonometric functions of angles outside the first quadrant using reference angles. All along the curve, any two points with opposite x -values have the same function value. Licenses and Attributions. An illustration of which trigonometric functions are positive in each of the quadrants.

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Skip to main content. An angle in the first quadrant is its own reference angle. Find the value of cosine or sine at the reference angle by looking at quadrant one of the unit circle. The first coordinate in each ordered pair is the value of cosine at the given angle measure, while the second coordinate in each ordered pair is the value of sine at the given angle measure. Lesson looked at the unit circle. One method to solve this problem is to sketch a right triangle in the specified quadrant with an acute angle at the origin and right angle on the x -axis. Summary: In this section, you will: Evaluate trigonometric functions of any angle. By combining the ideas of the unit circle and right triangles, the location of any capsule on the Eye can be described with trigonometry. A right triangle can be drawn to the point where one acute angle is at the point, the other acute angle is at the origin, and the right angle is on the x -axis. The secant of an angle is the same as the secant of its opposite. Pick a point on the circle.