Solved: Trigonometric Identities Problem

If you’re taking a geometry or trigonometry class, you’ll most likely learn trigonometric identities. 

In simple terms, trigonometric identities are equations that are related to different trigonometric functions and are true for any value of the variable. It could be hard since there are numerous trigonometry identities.

In this post you’re going to how to solve one of the top asked examples of trigonometric identities.

Solved: Trigonometric Identities Problem

Find the exact value of:

In this problem we’re asked to find the exact value of the given expression. Right away, the given expression should remind us of the sum and difference identities for sine and cosine shown below:

For our expression, notice A is equal to pi divided by 3 and B is equal to pi divided by 2. Due to the fact we have a sum, the expression is equal to the sine of the sum of the angles.

1) First, we have to find the sum of the angles and then determine the sine function value.

To add the fractions we have to find a common denominator. In this case, the least common denominator is a least common multiple of 2 and 3, which is 6. Therefore, we multiply the numerator and denominator of pi over 3 by 2, and we multiply the numerator and denominator of pi over 2 by 3:

2) Now we know:


We need to find the sine function value, which we can do using a reference triangle, or the unit circle. To use a reference triangle, we first sketch 5/6 pi radians in standard position. 

The initial side is along the positive x-axis, and then we rotate 5 pi divided by 6 radians counterclockwise, which is 1/6 pi, or pi divided by 6 radians short of 1/2 a rotation, which would be approximately here. this angle here is 5 pi divided by 6 radians, and the reference angle is the angle formed with the terminal side and this negative x-axis, which is pi over 6 radians:

3) Let’s sketch the reference triangle. We should recognize this as a 30-60-90 reference triangle, where we can label the short leg 1, the hypotenuse 2, and the longer leg square root of 3.

However, we are in the second quadrant where x is negative, and therefore, the square root of 3 is negative. We can use this reference triangle to determine the sine function value, where the sine of 5 pi divided by 6 radians is equal to the ratio of the opposite side to the hypotenuse, which we can see is equal to 1 divided by 2, or 1/2. So, the exact value of the given expression is equal to 1/2.

4) Let’s also find this sine function value on the unit circle. The terminal side of 5 pi divided by 6 radians intersects the unit circle. On the unit circle, remember sine theta is equal to y, and notice how the y-coordinate is equal to 1/2. 


Trig identities can be overwhelming- But that’s where a high-quality textbook survival guide comes in.With it, you’ll not only be able to get the right answers, but understand concepts to prepare for the exams quickly without stress.

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