Amy has a master's degree in secondary education and has taught math at a public charter high school.
Trigonometry is full of identities, a set of which are called the reciprocal identities. Watch this video lesson to learn what they are and how you can use them to help you solve problems.
The Reciprocal Identities
In this video lesson, we talk about the reciprocal identities of trigonometry. What are they? They are the definitions of our trig functions in terms of another trig function. They tell us how the trig functions are related to each other. They also tell us which trig functions are reciprocals of each other.
Remember in math a reciprocal of a number is 1 divided by that number? For example, the reciprocal of 5 is 1/5. Well, in trigonometry, the reciprocal of a trig function is 1 divided by another trig function. What are they? Here are six of them:
This trig function
Is equal to
Look carefully and you will spot corresponding pairs. For example, because the sine function is equal to 1 over the cosecant function, we also have the cosecant function is equal to 1 over the sine function. Notice how the two corresponding trig functions have simply switched places? We have six statements, so you will see three corresponding pairs. Can you spot them?
Now we know what the reciprocal identities are. So, what can we do with them? We can use these identities, or true statements, to help us simplify trig problems. We do this by substituting our definitions into the problem where we can to help us simplify and then solve the problem.
For example, the problem sin^2 (pi / 2) * csc (pi / 2) might seem difficult at first. However, we see that we have a sine function, as well as a cosecant function. We know that they are reciprocal functions. We can actually substitute 1 over sine in for the cosecant.
After we do that, our function becomes very simple. With the reciprocal, one of our sine functions can be canceled and we are left with just a single sine of pi/2. pi/2 is a radian measure. We can use either our unit circle to find our answer or we can input this into a calculator, making sure our calculator is set to radians. Doing this we see that we get a nice answer of 1:
A good rule of thumb to follow to make your problem solving easier is to rewrite your functions into sine, cosine, and tangent. So, if you see two functions that are reciprocals of each other, rewrite the one that isn't the sine, cosine, or tangent function.
Let's look at a couple more examples to see how we can use our reciprocal identities. Look at this problem: Simplify cos (theta) * sec (theta) * tan (theta) * cot (theta).
At first look, we might wonder how we could possibly simplify this? We have four functions all being multiplied together. How can we possibly cancel anything out?
Not to worry, our reciprocal identities are here to help us out. We have a cosine and a secant. They are reciprocals. So, we can write our secant as 1 over cosine; the same with tangent and cotangent. We can write our cotangent as 1 over tangent.
Over 79,000 lessons in all major subjects
Get access risk-free for 30 days,
just create an account.
What happens after we do that? We can start canceling out terms. What are we left with? Why, a big, simple 1! Yay! That was very cool!
Let's look at one more:
How can we simplify this? We see that we have a sine function, a secant function, and a cosine function. Which ones are reciprocals of each other?
The secant and cosine. We can rewrite the secant function as 1 over cosine. But wait, our secant is already in the denominator. What do we do?
Easy, we rewrite the cosine in the numerator this time. Why? Because 1 over secant is equal to cosine in the numerator. So, now we have a cosine in the numerator and a cosine in the denominator. We can cancel these out. What are we left with?
The sine of theta. And at this point, we can't do anything more, so we are done. Our problem simplified all the way down to sine of theta:
Let's review what we've learned now. We learned that our reciprocal identities are the definitions of our trig functions in terms of another trig function. We have a total of six reciprocal identities. The identities actually pair up into three corresponding pairs of reciprocals. Here is the table of all the reciprocal identities:
This trig function
Is equal to
We use these reciprocal trig identities to help us simplify trig problems into simpler ones that we can easily solve. A good rule of thumb to follow when simplifying our problems with these reciprocal identities is to stick to rewriting things in terms of cosine, sine, and tangent. After rewriting our problem with the aid of our reciprocal identities, we can then cancel terms out to find our simplified expression.
Following this lesson, you'll be able to:
Define reciprocal identities and identify the six reciprocal identities
Explain how these reciprocal identities can be used to make problem solving easier
Did you know… We have over 200 college
courses that prepare you to earn
credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the
first two years of college and save thousands off your degree. Anyone can earn
credit-by-exam regardless of age or education level.