We label the hypotenuse with the symbol h. There is a side opposite the angle c which we label o for "opposite". The remaining side we label a for "adjacent". The angle c is formed by the intersection of the hypotenuse h and the adjacent side a. We are interested in the relations between the sides and the angles of the right triangle.
Let us start with some definitions. We will call the ratio of the opposite side of a right triangle to the hypotenuse the sine and give it the symbol sin. The ratio of the adjacent side of a right triangle to the hypotenuse is called the cosine and given the symbol cos.
Finally, the ratio of the opposite side to the adjacent side is called the tangent and given the symbol tan. We claim that the value of each ratio depends only on the value of the angle c formed by the adjacent and the hypotenuse. To demonstrate this fact, let's study the three figures in the middle of the page. In this example, we have an 8 foot ladder that we are going to lean against a wall.
You can also see Graphs of Sine, Cosine and Tangent. To complete the picture, there are 3 other functions where we divide one side by another, but they are not so commonly used. They are equal to 1 divided by cos , 1 divided by sin , and 1 divided by tan :. Adjacent is always next to the angle And Opposite is opposite the angle. Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own. Unlimited random practice problems and answers with built-in Step-by-step solutions.
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Wolfram Language » Knowledge-based programming for everyone. Terms of Use. Discrete Versions of Continuous Functions. Tangent The tangent function is defined by. Substitute these values into the definitions for the six ratios. Notice that the values of sine and cosine are between 0 and 1. You found them by dividing the length of a leg by the hypotenuse. The hypotenuse is the longest side, so the numerator is less than the denominator.
That means the output of the sine or cosine function is always less than 1. Keep in mind that the opposite side for one acute angle is the adjacent side of the other acute angle. In the example above, side EF was the opposite side for angle D.
Determine the six trigonometric ratios for angle E in the right triangle below. This is the same triangle that you saw in the previous example, so the hypotenuse is the same. The difference is that you are looking at the triangle from the perspective of angle E instead of angle D. So the opposite and adjacent sides switch places.
That is, is adjacent to angle E and is opposite angle E. Substitute the new values into the definitions for the six ratios. If you compare the answers to the last two examples, you will see the following:. These two trigonometric functions are equal because the opposite side to angle D which is 4 is the adjacent side to angle E. Because they are the two acute angles in a right triangle, D and E are complementary. That is:. Substitute this into the equation above:.
Again, the reason these two functions are equal is that the opposite side to one acute angle is the adjacent side to the other acute angle. This is true in any right triangle.
So if A is any acute angle, it is always true that:. Comparing more answers from the last two examples, you can find these relationships:. You get these equalities because 1 the adjacent side to angle D is 3, while this is the opposite side to angle E , and 2 the opposite side to angle D is 4, while this is the adjacent side to angle E.
These are examples of the general relationship we have stated: the opposite side to one acute angle is the adjacent side to the other acute angle. Using the same reasoning as above, if A is any acute angle, it is always true that:.
An equation, such as any of the three above, that is true for any value of the variable is called an identity. Note the full names of these functions: sine and co sine, secant and co secant, tangent and co tangent. These pairs are referred to as cofunctions. The angles A and are complementary. In other words, the cofunctions of any pair of complementary angles are equal. You can use these relationships to find values of trigonometric functions from values of other functions without drawing a triangle.
Note that you can replace A and by B and. The different letter will not change the relationship, because these angles are still complementary. Since A and B are the acute angles in a right triangle, they are complementary angles. Substitute for B. Use the identity the cofunctions are equal. Substitute the given value. Substitute for A. The cofunctions of any pair of complementary angles are equal. What are the values of and?
You probably used the acute angle W , and found. Remember that you get different ratios for the two acute angles, so pay careful attention to which angle you are using. The correct answer is C. You may have used the acute angle W and also switched cosine and cosecant. Using the definition of cosine,. Using the definition of cosecant,. It looks like you switched the values of cosine and cosecant.
The names are very similar, so be careful to use the right definition. Relationships Among the Trigonometric Functions. The six ratios or functions are usually thought of as two groups of three functions. The first group is:. One way to remember these three definitions is with a memory device that uses the first letter of each word.
The definition of sine is represented by soh s ine equals o pposite over h ypotenuse. Likewise, the definition of cosine is represented by cah c osine equals a djacent over h ypotenuse , and the definition of tangent is represented by toa t angent equals o pposite over a djacent. Putting these together gives you sohcahtoa. That is, cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent. Writing this gives three more identities:.
If you remember sohcahtoa plus these three identities, you can find the values of any trigonometric functions, as seen in the following example. For acute angle A , and. Find the values of the other four trigonometric ratios for angle A. The definition of sine tells you that. A triangle with and will have this ratio. You also know that. You are given , so.
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