Question Video: Finding the Unknown Length in a Right Triangle Using Trigonometry Where the Unknown Is on the Top of the Fraction Mathematics

Find 𝑥 in the given figure. Give your answer to two decimal places.

02:17

Video Transcript

Find 𝑥 in the given figure. Give your answer to two decimal places.

In this right triangle, we have been given the length of one side and the measure of one of the other angles. We want to calculate the length of another side. We’ll begin by labeling the size of the triangle in relation to the angle of 68 degrees. The side directly opposite the right angle is always the hypotenuse, which we’ll abbreviate to H. The side directly opposite the angle of 68 degrees is the opposite side, which we’ll abbreviate to O. And finally, the side between the right angle and the angle of 68 degrees is the adjacent, which we’ll abbreviate to A. We can use the acronym SOHCAHTOA to help us decide which of the trigonometric ratios we need to use in this question.

The side we wish to calculate is the opposite and the side whose length we know is the adjacent, which tells us we’re going to be using the tangent ratio. Let’s recall its definition. In a right triangle, the tan of an angle 𝜃 is equal to the length of the opposite side divided by the length of the adjacent side. Now we substitute the value of 𝜃 and the values or expressions for the opposite and adjacent in this triangle. 𝜃 is 68 degrees. The opposite we don’t know, but we have the expression 𝑥. And the adjacent is 11 units. So we have the equation tan of 68 degrees is equal to 𝑥 over 11.

To solve this equation for 𝑥, we need to multiply both sides by 11. This gives 11 multiplied by tan of 68 degrees is equal to 𝑥. Or we could just write this as 11 tan 68 degrees. We don’t need the multiplication sign. We can now evaluate this on our calculators, making sure they are in degree mode. And it gives 27.225. The question specifies that we should give our answer to two decimal places. So rounding our answer, we have 27.23. No units were given in the question, so our answer is simply 27.23 length units.

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