We now enter into the proof portion of trigonometry. Starting with the basic definitions of sine, cosine, and tangent, identities (or fundamental trigonometric equations) emerge. Students will learn how to prove certain identities, using other identities and definitions. Finally, students will be able solve trigonometric equations for theta, also using identities and definitions.
- use the fundamental identities to prove other identities.
- apply the fundamental identities to values of and show that they are true.
In Chapter 1, the three fundamental trigonometric functions sine, cosine and tangent were introduced. All three functions can be defined in terms of a right triangle or the unit circle.
The Quotient Identity is . We see that this is true because tangent is equal to , in the unit circle. We know that is equal to the sine values of and is equal to the cosine values of . Substituting for and for and we have a new identity.
Example 1: Use to show that holds true.
Solution: Plugging in , we have: . Then, substitute each function with its actual value and simplify both sides.
and we know that tan , so this is true.
Example 2: Show that tan is undefined using the Quotient Identity.
Solution: , because you cannot divide by zero, the tangent at is undefined.
Chapter 1 also introduced us to the idea that the three fundamental reciprocal trigonometric functions are cosecant (csc), secant (sec) and cotangent (cot) and are defined as:
If we apply the Quotient Identity to the reciprocal of tangent, an additional quotient is created:
Example 3: Prove
Solution: First, you should change everything into sine and cosine. Feel free to work from either side, as long as the end result from both sides ends up being the same.
Here, we end up with the Quotient Identity, which we know is true. Therefore, this identity is also true and we are finished.
Example 4: Given and is in the fourth quadrant, find .
Solution: Secant is the reciprocal of cosine, so we need to find the adjacent side. We are given the opposite side, and the hypotenuse, . Because is in the fourth quadrant, cosine will be positive. From the Pythagorean Theorem, the third side is:
From this we can now find . Since secant is the reciprocal of cosine, , or .
Using the fundamental trig functions, sine and cosine and some basic algebra can reveal some interesting trigonometric relationships. Note when a trig function such as is multiplied by itself, the mathematical convention is to write it as . ( can be interpreted as the sine of the square of the angle, and is therefore avoided.)
Using the Pythagorean Theorem for the triangle above:
Then, divide both sides by . So, because also equals . This is known as the Trigonometric Pythagorean Theorem or the Pythagorean Identity and is written . Alternative forms of the Theorem are: and . The second form is found by taking the original form and dividing each of the terms by , while the third form is found by dividing all the terms of the first by .
Example 5: Use to show that holds true.
Solution: Plug in and find the values of and .
Even and Odd Identities
Functions are even or odd depending on how the end behavior of the graphical representation looks. For example, is considered an even function because the ends of the parabola both point in the same direction and the parabola is symmetric about the axis. is considered an odd function for the opposite reason. The ends of a cubic function point in opposite directions and therefore the parabola is not symmetric about the axis. What about the trig functions? They do not have exponents to give us the even or odd clue (when the degree is even, a function is even, when the degree is odd, a function is odd).
Let’s consider sine. Start with . Will it equal or ? Plug in a couple of values to see.
From this we see that sine is odd. Therefore, , for any value of . For cosine, we will plug in a couple of values to determine if it’s even or odd.
This tells us that the cosine is even. Therefore, , for any value of . The other four trigonometric functions are as follows:
Notice that cosecant is odd like sine and secant is even like cosine.
Example 6: If and , find .
Solution: We know that sine is odd. Cosine is even, so . Tangent is odd, so . Therefore, sine is positive and .
Recall that two angles are complementary if their sum is . In every triangle, the sum of the interior angles is and the right angle has a measure of . Therefore, the two remaining acute angles of the triangle have a sum equal to , and are complementary. Let’s explore this concept to identify the relationship between a function of one angle and the function of its complement in any right triangle, or the cofunction identities. A cofunction is a pair of trigonometric functions that are equal when the variable in one function is the complement in the other.
In is a right angle, and are complementary.
Chapter 1 introduced the cofunction identities (section 1.8) and because and are complementary, it was found that and . For each of the above . To generalize, and and and .
The following graph represents two complete cycles of and .
Notice that a phase shift of on , would make these graphs exactly the same. These cofunction identities hold true for all real numbers for which both sides of the equation are defined.
Example 7: Use the cofunction identities to evaluate each of the following expressions:
a. If determine
b. If determine .
Example 8: Show is true.
Solution: Using the identities we have derived in this section, , and we know that cosine is an even function so . Therefore, each side is equal to and thus equal to each other.
Points to Consider
- Why do you think secant is even like cosine?
- How could you show that tangent is odd?
- Use the Quotient Identity to show that the tan is undefined.
- If , find .
- If and , find .
- Simplify .
- Verify using:
- the sides , and of a right triangle, in the first quadrant
- the ratios from a triangle
- Prove using the Pythagorean Identity
- If and , find .
- Simplify using the trig identities
- Rewrite so that it is only in terms of cosine. Simplify completely.
- Prove that tangent is an odd function.
, you cannot divide by zero, therefore is undefined.
- If , then, by the cofunction identities, . Because sine is odd, .
- If , then . Because , cosine is also negative, so .
- Use the reciprocal and cofunction identities to simplify
- (a) Using the sides 5, 12, and 13 and in the first quadrant, it doesn’t really matter which is cosine or sine. So, becomes . Simplifying, we get: , and finally . (b) becomes . Simplifying we get: and .
- To prove , first use and change .
- If and , then and . Therefore .
- (a) Factor using the difference of squares. (b)
- You will need to factor and use the identity.
- To rewrite so it is only in terms of cosine, start with changing secant to cosine. Multiply by the reciprocal
- The easiest way to prove that tangent is odd to break it down, using the Quotient Identity.