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

For the given figure, find the measure of โˆ ๐ต๐ด๐ถ, in degrees, to two decimal places.

03:22

Video Transcript

For the given figure, find the measure of angle ๐ต๐ด๐ถ in degrees to two decimal places.

Looking at the diagram, we can see that we have a right-angled triangle in which the lengths of two of the sides have been given. Weโ€™re asked to find the measure of angle ๐ต๐ด๐ถ, which is the angle formed by moving from ๐ต to ๐ด to ๐ถ, so the angle marked in orange. As this is a right-angled triangle, weโ€™re going to use trigonometry in order to calculate the size of this angle.

We begin by labelling the three sides of the triangle with their names in relation to the angle ๐ต๐ด๐ถ. The hypotenuse is always the side opposite the right angle. The side opposite the angle weโ€™re looking to calculate is called the opposite. So here, itโ€™s side ๐ต๐ถ. The third side, which sits between the right angle and the angle of interest, is called the adjacent. So here, this is side ๐ด๐ต.

Next we, need to consider which of the three trigonometric ratios โ€” sine, cosine, or tangent โ€” we need to use in this question. Remember, you can use the acronym SOHCAHTOA to help with this, where S, C, and T stand for sine, cosine, and tangent and O, A, and H stand for opposite, adjacent, and hypotenuse.

In this question, the two sides whose lengths weโ€™ve been given are the opposite and the adjacent sides. Therefore, it is the tan ratio that we need to use. So tan of angle ๐ต๐ด๐ถ is equal to the opposite divided by the adjacent. Next, we need to substitute the values for the opposite and the adjacent in this question. That is seven for the opposite and five for the adjacent.

So now we have that tan of angle ๐ต๐ด๐ถ is equal to seven over five. We donโ€™t want to just know the value of the ratio, tan ๐ต๐ด๐ถ. We want to work out angle ๐ต๐ด๐ถ itself. So now we need to use the inverse tan function. This is the function that allows us to work backwards from knowing the value of the tan ratio, seven over five, to finding the size of the angle that is associated with this ratio.

So we have that angle ๐ต๐ด๐ถ is equal to tan inverse of seven over five. Remember, this is an inverse function and shouldnโ€™t be confused with a reciprocal function, one divided by tan. They donโ€™t mean the same thing. Now you need to use your calculator to evaluate this. And normally the tan inverse function is situated above the tan button. So you need to press shift in order to get to it.

Evaluating this tell that angle ๐ต๐ด๐ถ is equal to 54.46232 and the decimal continues. Remember, weโ€™ve been asked to find the measure of this angle to two decimal places. So we now need to round this value. So we have that the measure of angle ๐ต๐ด๐ถ to two decimal places is 54.46 degrees.

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