For the following exercises, draw the angle provided in standard position on the Cartesian plane. Find the linear speed of a point on the equator of the earth if the earth has a radius of 3, miles and the earth rotates on its axis every 24 hours. Express answer in miles per hour. Round to the nearest hundredth. A car wheel with a diameter of 18 inches spins at the rate of 10 revolutions per second. For the following exercises, use the given information to find the lengths of the other two sides of the right triangle.
For the following exercises, use Figure to evaluate each trigonometric function. For the following exercises, solve for the unknown sides of the given triangle.
Find the answer to four decimal places. The angle of elevation to the top of a building in Baltimore is found to be 4 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building. For the following exercises, use reference angles to evaluate the given expression.
A carnival has a Ferris wheel with a diameter of 80 feet. The time for the Ferris wheel to make one revolution is 75 seconds. What is the linear speed in feet per second of a point on the Ferris wheel?
What is the angular speed in radians per second? The angle of elevation to the top of a building in Chicago is found to be 9 degrees from the ground at a distance of feet from the base of the building.
Privacy Policy. Skip to main content. Search for:. Use properties of even and odd trigonometric functions. Recognize and use fundamental identities. Evaluate trigonometric functions with a calculator. Figure 1. Using Reference Angles to Evaluate Tangent, Secant, Cosecant, and Cotangent We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions.
How To Given an angle not in the first quadrant, use reference angles to find all six trigonometric functions. Measure the angle formed by the terminal side of the given angle and the horizontal axis. This is the reference angle. Evaluate the function at the reference angle.
Observe the quadrant where the terminal side of the original angle is located. Based on the quadrant, determine whether the output is positive or negative. Using Even and Odd Trigonometric Functions 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.
Figure 7. Show Solution Secant is an even function. Recognizing and Using Fundamental Identities We have explored a number of properties of trigonometric functions. Fundamental Identities We can derive some useful identities from the six trigonometric functions. Show Solution Because we know the sine and cosine values for these angles, we can use identities to evaluate the other functions.
Show Solution We can simplify this by rewriting both functions in terms of sine and cosine. Figure 8. Figure 9. Figure Evaluating Trigonometric Functions with a Calculator We have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them as reference angles for angles in other quadrants. How To Given an angle measure in radians, use a scientific calculator to find the cosecant. If the calculator has degree mode and radian mode, set it to radian mode.
Press the SIN key. If the graphing utility has degree mode and radian mode, set it to radian mode. Enter the value of the angle inside parentheses. Key Concepts The tangent of an angle is the ratio of the y -value to the x -value of the corresponding point on the unit circle.
The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function. The six trigonometric functions can be found from a point on the unit circle. See Figure. Trigonometric functions can also be found from an angle. Trigonometric functions of angles outside the first quadrant can be determined using reference angles.
Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd. Even and odd properties can be used to evaluate trigonometric functions.
The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine. Identities can be used to evaluate trigonometric functions. See Figure and Figure. Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities. The trigonometric functions repeat at regular intervals. To evaluate trigonometric functions of other angles, we can use a calculator or computer software.
Describe the secant function. Algebraic For the following exercises, find the exact value of each expression. Show Solution 1. Show Solution 2. Show Solution —1. Show Solution Show Solution —2. Show Solution 3. Graphical For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions. Technology For the following exercises, use a graphing calculator to evaluate to three decimal places.
Show Solution —0. Extensions For the following exercises, use identities to evaluate the expression. Show Solution even. Show Solution 7. The first function we will define is the tangent. The tangent of an angle is the ratio of the y -value to the x -value of the corresponding point on the unit circle. We can use these values and the definitions of tangent, secant, cosecant, and cotangent as functions of sine and cosine to find the remaining function values.
Because we know the sine and cosine values for the common first-quadrant angles, we can find the other function values for those angles as well by setting x x equal to the cosine and y y equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent.
We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the same: Find the reference angle formed by the terminal side of the given angle with the horizontal axis.
The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x - and y -values in the original quadrant. HOWTO: Given an angle not in the first quadrant, use reference angles to find all six trigonometric functions.
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. As it turns out, there is an important difference among the functions in this regard. The graph of the function is symmetrical about the y -axis. All along the curve, any two points with opposite x -values have the same function value. The graph is not symmetrical about the y -axis. All along the graph, any two points with opposite x -values also have opposite y -values.
The sine function , then, is an odd function. We can test each of the six trigonometric functions in this fashion. Secant is an even function.
The secant of an angle is the same as the secant of its opposite. We have explored a number of properties of trigonometric functions. Now, we can take the relationships a step further, and derive some fundamental identities. Identities are statements that are true for all values of the input on which they are defined. Usually, identities can be derived from definitions and relationships we already know.
For example, the Pythagorean Identity we learned earlier was derived from the Pythagorean Theorem and the definitions of sine and cosine. We can derive some useful identities from the six trigonometric functions. The other four trigonometric functions can be related back to the sine and cosine functions using these basic relationships:.
Because we know the sine and cosine values for these angles, we can use identities to evaluate the other functions. We can simplify this by rewriting both functions in terms of sine and cosine. The sign of the sine depends on the y -values in the quadrant where the angle is located.
The remaining functions can be calculated using identities relating them to sine and cosine. As we discussed in the chapter opening, a function that repeats its values in regular intervals is known as a periodic function. The trigonometric functions are periodic.
And for tangent and cotangent, only a half a revolution will result in the same outputs. Other functions can also be periodic. For example, the lengths of months repeat every four years. This pattern repeats over and over through time. In other words, every four years, February is guaranteed to have the same number of days as it did 4 years earlier. The positive number 4 is the smallest positive number that satisfies this condition and is called the period.
A period is the shortest interval over which a function completes one full cycle—in this example, the period is 4 and represents the time it takes for us to be certain February has the same number of days.
We have learned how to evaluate the six trigonometric functions for the common first-quadrant angles and to use them as reference angles for angles in other quadrants. To evaluate trigonometric functions of other angles, we use a scientific or graphing calculator or computer software. If the calculator has a degree mode and a radian mode, confirm the correct mode is chosen before making a calculation.
Evaluating a tangent function with a scientific calculator as opposed to a graphing calculator or computer algebra system is like evaluating a sine or cosine: Enter the value and press the TAN key.
0コメント