Find the measure of angle 𝜃 in degrees to two decimal places.
This question shows a right-angled triangle, in which we’re given the lengths of two of the sides and asked to calculate the measure of one of the other angles. We can, therefore, answer this question using right angle trigonometry. The first step for any problem involving trigonometry is to label the three sides of the triangle in relation to the angle we’re interested in, so that’s the angle 𝜃.
The hypotenuse and the longest side of the triangle is always the side directly opposite the right angle. The side directly opposite the angle we’re interested in is called the opposite. The third side of the triangle which is between the angle we’re interested in and the right angle is called the adjacent.
Next, we need to decide which of the three trigonometric ratios sine, cosine, or tan we need to use in this question. And to do so, we can recall the acronym SOHCAHTOA. Here, S, C, and T stand for sine, cosine, and tan. And O, A, and H stand for opposite, adjacent, and hypotenuse. The two sides of the triangle that we’ve been given are the adjacent and the hypotenuse. So this is the CAH part of SOHCAHTOA, which means it’s the cosine ratio we’re going to use in this question. Let’s recall its definition.
The cosine ratio of an angle 𝜃 is equal to the adjacent divided by the hypotenuse. For any triangle with a given angle 𝜃, the ratio between these two sides will always be the same. In this question, the adjacent side is five units and the hypotenuse is eight units. So we can substitute these values into the ratio. We, therefore, have that cos of our angle 𝜃 is equal to five-eighths.
To find our angle 𝜃, we need to work backwards from knowing what its cosine ratio is to work out what the angle itself is. And to do this, we need to use the inverse cosine function. The notation for this is cos and then a superscript negative one. And it means which angle belongs with the cosine ratio of five-eighths. You can evaluate this using your calculator. The inverse cosine function is usually located above the cos button. And you need to press shift in order to get to it.
We were asked for the measure of angle 𝜃 measured in degrees. So you need to make sure that your calculator is in the correct mode before performing any trigonometry. If you type this correctly into your calculator, you should get 51.317812.
The question asked for the answer to two decimal places. So the final step is just to round our answer. The third decimal place is a seven. So we round up. And we have that the measure of angle 𝜃 to two decimal places is 51.32 degrees.