Initial posting - July, 2011
Revision date - Fall 2017 Purpose: This lesson introduces the inverse sine function, which is used to find an angle measure in a given right triangle. Essential Question: Given a right triangle and three known side measurements, how can the inverse sine function be used to calculate the missing angle measures?

A Section of the Lesson Follows What would happen if all of the side measurements of a right triangle are given but the two acute angle measures are unknown? How would you find the two missing angle measures? Let’s revisit the mathematics in a right triangle. Using what you know about Pythagorean triples, let the side lengths of triangle PAQ be 30, 40 and 50 turtle steps. Since these lengths are the Pythagorean triple 3, 4, 5, times a factor of 10, these side measurements will form a right triangle.

Note that you do not know either the interior or the exterior angles at points P and Q. But, since this is a right triangle you can use the sine function to find the triangle’s interior angles. Recall that the sine function uses three values, one of the non-right angles, the opposite side length and the hypotenuse. The set up looks like this: sin(m(∠P)) = 40/50 In words, this reads as “the sine of the measure of angle P equals 40 divided by 50.”

To “undo” the sine function I must take the inverse. The inverse symbol for the sine function is written as “sine to the negative one power” or sin-1.

The next step looks like this: sin-1 [sin(m(∠P))] = sin-1(40/50)

When I take the inverse of the sine function on the left side of this equation, I must do so on the other side of the equation!

The next step looks like this: sin-1 [sin(m(∠P))] = sin-1(40/50)

Simplify the equation: sin-1 [sin(m(∠P))] = sin-1(40/50) m(∠P) = sin-1 (40/50) m(∠P) ≈ 53.1 degrees (tenths is sufficient here)

Repeat these steps to find the measure of angle Q. sin(m(∠Q)) = 30/50 sin-1 (sin(m(∠Q))) = sin-1 (30/50) m(∠Q) = sin-1 (30/50) ≈ 36.9 degrees

A good check of the calculations would be to add 36.9° and 53.1°, to confirm that the total is 90 degrees. You may have noticed that subtracting 53.1° from 90 degrees could have been done to find the measurement of angle Q, because the sum of the angle measures for any triangle has to be 180 degrees. Return to the diagram and write in the angle measurements. To complete the Logo code, find the angle of turn for the turtle as you trace the path from point A clockwise around the triangle. Greek letters are often used in mathematics as a variable for an unknown angle measurement. In the figure below the letters alpha (α) and beta (β) are used to indicate measures of the exterior angles at points P and Q, respectively.

## Lesson #11 - Inverse Sine Function

Initial posting - July, 2011Revision date - Fall 2017

Purpose:This lesson introduces the inverse sine function, which is used to find an angle measure in a given right triangle.Essential Question:Given a right triangle and three known side measurements, how can the inverse sine function be used to calculate the missing angle measures?A Section of the Lesson Follows

What would happen if all of the side measurements of a right triangle are given but the two acute angle measures are unknown? How would you find the two missing angle measures?

Let’s revisit the mathematics in a right triangle. Using what you know about Pythagorean triples, let the side lengths of triangle PAQ be 30, 40 and 50 turtle steps. Since these lengths are the Pythagorean triple 3, 4, 5, times a factor of 10, these side measurements will form a right triangle.

Note that you do not know either the interior or the exterior angles at points P and Q. But, since this is a right triangle you can use the sine function to find the triangle’s interior angles. Recall that the sine function uses three values, one of the non-right angles, the opposite side length and the hypotenuse. The set up looks like this:

sin(m(∠P)) = 40/50 In words, this reads as “the sine of the measure of angle P equals 40 divided by 50.”

To “undo” the sine function I must

take the inverse. The inverse symbol for the sine function is written as “sine to the negative one power” or sin-1.The next step looks like this:

sin-1 [sin(m(∠P))] = sin-1(40/50)

When I

take the inverseof the sine function on the left side of this equation, I must do so onthe other sideof the equation!The next step looks like this: sin-1 [sin(m(∠P))] = sin-1(40/50)

Simplify the equation: sin-1 [sin(m(∠P))] = sin-1(40/50)

m(∠P) = sin-1 (40/50)

m(∠P) ≈ 53.1 degrees (tenths is sufficient here)

Repeat these steps to find the measure of angle Q.

sin(m(∠Q)) = 30/50

sin-1 (sin(m(∠Q))) = sin-1 (30/50)

m(∠Q) = sin-1 (30/50) ≈ 36.9 degrees

A good check of the calculations would be to add 36.9° and 53.1°, to confirm that the total is 90 degrees. You may have noticed that subtracting 53.1° from 90 degrees could have been done to find the measurement of angle Q, because the sum of the angle measures for any triangle has to be 180 degrees.

Return to the diagram and write in the angle measurements. To complete the Logo code, find the angle of turn for the turtle as you trace the path from point A clockwise around the triangle.

Greek letters are often used in mathematics as a variable for an unknown angle measurement. In the figure below the letters alpha (α) and beta (β) are used to indicate measures of the exterior angles at points P and Q, respectively.