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.
In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent. Access these online resources for additional instruction and practice with other trigonometric functions.
Jay Abramson Arizona State University with contributing authors. Use reference angles to evaluate the trigonometric functions secant, tangent, and cotangent. Use properties of even and odd trigonometric functions. Recognize and use fundamental identities. Evaluate trigonometric functions with a calculator. The secant function is the reciprocal of the cosine function.
The cotangent function is the reciprocal of the tangent function. 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.
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.
Solution Secant is an even function. Recognizing and Using Fundamental Identities We have explored a number of properties of trigonometric functions. Feb 28, Mar 1, Hope It Helps. Mar 10, Related questions How do you use the fundamental trigonometric identities to determine the simplified form of the How do you apply the fundamental identities to values of theta and show that they are true?
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. In that case, the function must be evaluated as the reciprocal of a sine, cosine, or tangent.
Given an angle measure in radians, use a scientific calculator to find the cosecant. Access these online resources for additional instruction and practice with other trigonometric functions. If so, where? Explain your reasoning. For any angle in quadrant II, if you knew the sine of the angle, how could you determine the cosine of the angle?
For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions. For the following exercises, use a graphing calculator to evaluate to three decimal places. Use the equation to find how many hours of sunlight there are on February 10, the 42 nd day of the year.
State the period of the function. Use the equation to find how many hours of sunlight there are on September 24, the th day of the year. 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.
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