Question Video: Finding the Unknown Angle in a Right Triangle Using Trigonometry | Nagwa Question Video: Finding the Unknown Angle in a Right Triangle Using Trigonometry | Nagwa

Question Video: Finding the Unknown Angle in a Right Triangle Using Trigonometry Mathematics • Third Year of Preparatory School

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For the given figure, find the measure of angle ๐œƒ, in degrees, to two decimal places.

02:15

Video Transcript

For the given figure, find the measure of angle ๐œƒ in degrees to two decimal places.

Looking at the figure, we can see that we have a right triangle, in which ๐œƒ represents the measure of one of the non-right angles. Weโ€™ve also been given the lengths of two of the sides of the right triangle. They are three and eight units. We can therefore approach this problem using trigonometry.

Our first step in any problem involving trigonometry is to label the three sides of the triangle in relation to the angle ๐œƒ. The side directly opposite the right angle is the hypotenuse, which weโ€™ll abbreviate to H. The side directly opposite the angle ๐œƒ is called the opposite, abbreviated to O. And the side between the angle ๐œƒ and the right angle is the adjacent, which weโ€™ll abbreviate to A.

Weโ€™ll now recall the acronym SOHCAHTOA to help us decide which of the three trigonometric ratios we need to use in this problem. The two side lengths weโ€™ve been given are the adjacent and the hypotenuse. So weโ€™re going to be using the cosine ratio. The cos of an angle ๐œƒ is defined to be the length of the adjacent side divided by the length of the hypotenuse. Substituting the values for this triangle, we have that cos of ๐œƒ is equal to three over eight, or three-eighths.

Now we need to find the value of ๐œƒ, which means we need to apply the inverse cosine function. This is the function that essentially does the opposite of the cosine function. It says if cos of ๐œƒ is equal to three-eighths, then what is ๐œƒ? We have ๐œƒ is equal to the inverse cos of three-eighths.

We can then evaluate this on our calculators, making sure theyโ€™re in degree mode. To access the inverse cosine function, we usually need to press shift and then the cos button on our calculators. Evaluating gives 67.975. And then we round to two decimal places as specified in the question. So, by applying the inverse cosine function in this right triangle, we found that the measure of angle ๐œƒ to two decimal places is 67.98 degrees.

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