For the given figure, find the measure of angle 𝜃 in degrees to two decimal places.
As the triangle is right-angled, we can use the trigonometrical ratios sine, cosine, and tangent, shortened to sin, cos, and tan. In any right-angled triangle, sin 𝜃 is equal to the opposite divided by the hypotenuse, cos 𝜃 is equal to the adjacent side divided by the hypotenuse, and tan 𝜃 is equal to the opposite divided by the adjacent.
Our first step is to label the triangle. The longest side of a right-angled triangle is the hypotenuse. The other two sides are determined by which angle we’re dealing with: which one is opposite the angle 𝜃 and which side is adjacent to the angle 𝜃. Well, in this case, the side labelled three is adjacent to the angle 𝜃 and the other side is opposite the angle.
As our measurements are on the adjacent and the hypotenuse, we’re going to use the ratio cos 𝜃 equals the adjacent divided by the hypotenuse. Substituting the values into the formula gives us cos 𝜃 equals three divided by eight or three-eighths.
In order to calculate the angle, we need to use the inverse function or cos to the minus one. Therefore, 𝜃 equals inverse cos of three-eighths. Ensuring our calculator is in degree mode, we can type this in, giving us an answer 𝜃 equals 67.98 degrees. This value has been rounded to two decimal places.
The trigonometrical ratios can be used to find missing angles or missing sides in right-angled triangles. Whenever we wish to find a missing angle, we will have to use the inverse function: inverse sin, inverse cos, or inverse tan.