Inverse Sine

Part of the Right-Triangle Trigonometry section of Coddy's Trigonometry journey. Lesson 23 of 50.

So far the angle was known and a side was not. Turn it round: a right triangle has an opposite side of 5 and a hypotenuse of 13, and you want the angle θ. The ratio is easy to write, sin(theta) = 5/13, but which angle has a sine of 5/13?

$Right triangle with the angle θ, the opposite side 5 and the hypotenuse 13$

The inverse sine answers that question. arcsin(5/13), also written sin⁻¹(5/13), is the angle whose sine is 5/13: about 22.62°. The −1 is a name, not a power: sin⁻¹ x is an angle, never 1 divided by sin x.

On the board, after Sine ratio writes sin(theta) = 5/13, the chip Use arcsin: every angle in the domain writes theta = asin(5/13), and the round chip turns it into a number of degrees. The goal row says θ lies between 0° and 90°, because an acute angle of a right triangle always does.

Challenge

Triangle ABC has a right angle at C. For the angle θ at A, the opposite side is 7 and the hypotenuse is 11. Find θ in degrees, rounded to two decimal places. Then press Check.

Try it yourself

$Use the figure to find theta$