Video: Using Trigonometric Ratios to Find the Measures of Angles in Right-Angled Triangles

Kathryn Kingham

Find the value of 𝜃 giving the answer in radians to two decimal places.


Video Transcript

Find the value of 𝜃, giving the answer in radians to two decimal places.

The first thing we wanna notice is that this triangle sits inside a rectangle. And that confirms for us that this angle on the bottom is a right angle. Once we know that, we can use our trig ratios to find the angle measure that’s missing. If angle 𝜃 is our reference point, the side measuring 16 centimeters is the opposite side and the side measuring 20 centimeters is the hypotenuse.

Opposite over hypotenuse is the sine ratio. In our triangle, sine of 𝜃 equals 16 over 20. But how do we isolate this angle measure? We take the sine inverse. The sine inverse of sine 𝜃 is equal to 𝜃. And if we take the sine inverse on the left side of our equation, we have to take the sine inverse on the right side of our equation.

Our angle measure 𝜃 is equal to the sine inverse of 16 over 20. Now you’ll use some form of technology: a calculator, a computer, or maybe even a chart with listed sine ratios. If you’re using a calculator or a computer, you wanna make sure that it’s set to radians and then enter the sine inverse of 16 over 20. You’ll get an irrational number 0.927295 and on and on.

Rounding this to two decimal places, to the hundredths place, we see that the digit to the right of the hundredths place is a seven; it’s a larger than five. So we round the hundredths place up. The two becomes a three and everything to the left of the hundredths place stays the same.

Just to know, if your calculator or computer told you that the answer was 53 and 130 thousandths, you have it set to degrees. That is the sin inverse of 16 over 20 in degrees. And that’s not what we’re looking for here. The angle 𝜃 measured in radians is 0.93 radians.

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