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. Figure 6. We can test whether a trigonometric function is even or odd by drawing a unit circle with a positive and a negative angle, as in Figure.
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. 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 at the beginning of the chapter, 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. 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. 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.
Let us explore more about the even function in detail along with its graphical representation and properties. Let us first understand the meaning of even functions, algebraically. Now, let us see what this means. For an even function f x , if we plug in -x in place of x, then the value of f -x is equal to the value of f x.
Interestingly, the above functions have even powers. Determine the value of f -x and identify if it is an even function or not. Let us now see how an even function behaves graphically.
The above graph of an even function is symmetric with respect to the y-axis. In other words, the graph of an even function remains the same after reflection about the y-axis.
After understanding the even function meaning, we are going to explore its properties. The sum of 3 odd numbers is odd-true or false? Is the gcf of any two odd numbers is always odd true or false? The difference of two odd numbers is always odd true or false? Are all multiples of 3 odd true or false?
Is it true or false that the greatest common factor of any 2 odd numbers is odd? What is the difference between negative sine and cosine graphs and positive sine and cosine graphs? Is eighty-seven odd number true or false? Is true or false a heptagon has an odd numbers of sides? Is true or false a triangle has an odd numbers of sides? Is it true that if the GCF of any two odd numbers is always even true or false?
How Only odd numbers are prime numbers true or false? What is even and odd wave function? Is odd numbers cannot be composite true or false? Is the sum of an even number and an odd number is always even True or False? Only regular polygons with an odd number of sides are symmetrical true or false? If the GCF of an odd number and an even number is always even true or false? Is the square of a whole number is always an odd number true or false?
When you square a whole number you always get an odd number true or false? Is The smallest odd prime number is 1 true or false? The sum of three odd numbers is always odd true or false? Is all odd numbers greater than three are prime true or false?
When do you use even odd and neither functions? People also asked. Which natural phenomenon is the best example of periodic behavior?
View results. What is the minimum number of points required to mark all maximum minimum and zeros in a period of a sinusoid? The vertical of the function cosecant are determined by the points that are not in the domain?
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