5.2 Solve Applications: Sine, Cosine and Tangent Ratios.
5.2 Solve Applications: Sine, Cosine and Tangent Ratios.
Learning Objectives
By the end of this section, you will be able to:
- Find missing side of a right triangle using sine, cosine, or tangent ratios
- Find missing angle of a right triangle using sine, cosine, or tangent ratios
- Solve applications using right angle trigonometry
Sine, Cosine, and Tangent Ratios
We know that any right triangle has three sides and a right angle. The side opposite to the right angle is called the hypotenuse. The other two angles in a right triangle are acute angles (with a measure less than 90 degrees). One of those angles we call reference angle and we use θ (theta) to represent it.
The hypotenuse is always the longest side of a right triangle. The other two sides are called opposite side and adjacent side. The names of those sides depends on which of the two acute angles is being used as a reference angle.
In the right triangle each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.
EXAMPLE 1
Label the sides of the triangle and find the hypotenuse, opposite, and adjacent.
Solution
We labeled the sides with a lowercase letter to match the uppercase letter of the opposite vertex.
c is hypotenuse
a is opposite
b is adjacent
TRY IT 1.1
Label the sides of the triangle and find the hypotenuse, opposite and adjacent.
Answer
y is hypotenuse
z is opposite
x is adjacent
TRY IT 1.2
Label the sides of the triangle and find the hypotenuse, opposite and adjacent.
Answer
r is hypotenuse
t is opposite
s is adjacent
Trigonometric Ratios
Trigonometric ratios are the ratios of the sides in the right triangle. For any right triangle we can define three basic trigonometric ratios: sine, cosine, and tangent.
Let us refer to Figure 1 and define the three basic trigonometric ratios as:
Three Basic Trigonometric Ratios
- sine θ =
- cosine θ =
- tangent θ =
Where θ is the measure of a reference angle measured in degrees.
Very often we use the abbreviations for sine, cosine, and tangent ratios.
- sin θ =
- cos θ =
- tan θ =
Some people remember the definition of the trigonometric ratios as SOH CAH TOA.
Let’s use the from Example 1 to find the three ratios.
EXAMPLE 2
For the given triangle find the sine, cosine and tangent ratio.
Solution
First let’s label the sides of the triangle:
sin θ =
cos θ =
tan θ =
TRY IT 2.1
For the given triangle find the sine cosine and tangent ratio.
Answer
sin θ =
cos θ =
tan θ =
TRY IT 2.2
For the given triangle find the sine, cosine and tangent ratio.
Answer
sin θ =
cos θ =
tan θ =
In Example 2, our reference angles can be or . Using the definition of trigonometric ratios, we can write sinE= , cosE=, and tanE= .
When calculating we will usually round the ratios to four decimal places and at the end our final answer to one decimal place unless stated otherwise.
EXAMPLE 3
For the given triangle find the sine, cosine and tangent ratios. If necessary round to four decimal places.
Solution
We have two possible referenc