In this section, we will learn how to use right triangles to evaluate trigonometric functions. Before doing that, however, let’s review some key concepts of right triangles that can be helpful when solving.
The Pythagorean Theorem is helpful because if we know the lengths of any two sides of a right triangle, we can always find the length of the third side.
Pythagorean triples are integers \(a\text{,}\)\(b\text{,}\) and \(c\) that satisfy the Pythagorean Theorem. Activity 6.3.4 and Activity 6.3.5 highlight some of the most common types of Pythagorean triples: \(3-4-5\) and \(5-12-13\text{.}\) All triangles similar to the \(3-4-5\) triangle will also have side lengths that are multiples of \(3-4-5\) (like \(6-8-10\)). Similarly, this is true for all triangles similar to the \(5-12-13\) triangle.
Students should draw a right triangle with the hypotenuse labeled as \(11\) cm and one of the other angles (not the angle opposite the hypotenuse) is labeled \(60\)°.
Suppose you are asked to find one of the sides of the right triangle. What additional information would you need to find the length of another side of the triangle?
Students will probably notice that the Pythagorean Theorem is not helpful in this case because they only know the length of one side. This is a great opportunity to discuss how the Pythagorean Theorem is useful in finding side lengths when at least two sides are known.
When working with right triangles, it is often helpful to refer to specific angles and sides. One way this is done is by using letters, such as \(A\) and \(a\) to show that these are an angle-side pair because every angle has a side opposite the angle in a triangle. Note that the capital letter indicates the angle, and the lower case letter indicates the side.
When given an angle, all sides of a triangle can be labeled from that angle’s perspective. For example, from angle \(A\)’s perspective, the sides of a right triangle are labeled as:
Figure6.3.10.From the perspective of angle \(A\text{,}\) all sides of a right triangle can be labeled.
We will define trigonometric functions for a given angle \(\theta\) as ratios between the side lengths of a right triangle. The first three trigonometric functions we will discuss are the sine function, the cosine function, and the tangent function.
Notice that these are defined according to the sides of a triangle from the perspective of an angle \(\theta\text{.}\) This is why it is important to be able to label the triangle correctly!
The three trigonometric ratios we have worked with so far (sine, cosine, and tangent) are referred to as the basic trigonometric functions. There are three additional functions, cosecant, secant, and cotangent that are found by taking the reciprocal of the basic trigonometric functions. These three ratios are referred to as the reciprocal trigonometric functions and can be defined as follows:
Suppose you are given triangle \(ABC\text{,}\) where \(a=24\text{,}\)\(b=32\text{,}\) and \(c=40\text{,}\) with \(c\) being the hypotenuse of the triangle.
Students should draw a triangle where the side across from \(\theta\) is labeled "opposite", the side next to the angle \(\theta\) is labeled "adjacent", and the "hypotenuse" as the third side across from the \(90\)-degree angle.
Given that \(\cos\theta=\dfrac{12}{13}\text{,}\) we know that the side adjacent to \(\theta\) is \(12\) and the hypotenuse is \(13\text{.}\) Using the Pythagorean Theorem, the third side is \(5\text{.}\)
For each triangle given, determine which trigonometric function would be the most helpful in determining the length of the side of a triangle. Be sure to draw a picture of the triangle to help you determine the relationship between the given angle and sides. In each case assume angle \(C\) is the right angle.
In triangle \(ABC\text{,}\) angle \(A=24\)° and the hypotenuse has length \(14\text{.}\) Which trigonometric function will best help determine the length of side \(a\text{?}\)
Students should draw a right triangle where the \(31\)° is the angle formed with the ground, \(500\) meters as the side adjacent to the angle, and the height of the Eiffel Tower (side opposite \(31\)°) as \(x\text{.}\)
