3.5 Trigonometric Ratios

Trigonometry is an area of geometry that has its origin in the ancient study of the relationship of the sides and angles of a triangle. The Greek word trigon means “triangle,” and metron means “measure.” Applications of trigonometry are essential to many disciplines such as carpentry, engineering, surveying, and astronomy. If we want to figure out the height of a tree, do we have to climb the tree to figure it out? Fortunately, by using trigonometry, we can find out the measurements of tall objects, like a tree, without too much hassle. In this chapter, we will explore some basic properties of triangles and applications of the Pythagorean theorem and trigonometric ratios.

A right triangle has one 90° angle and two acute (less than 90°) angles. The 90° angle is often marked as shown in the diagram below.

A right triangle is shown. The right angle is marked with a box and labeled 90 degrees.

 

 

Many real-life problems can be represented using a right triangle. In this unit, we will use right-angle trigonometry to help us solve these problems.

Right Triangle Side Names

All right triangle have three sides and three angles. The side opposite to the right angle is called the hypotenuse. The other two sides are called the opposite side and adjacent side. The location of these sides depends on which of the two acute angles is the unknown angle you are trying to find in the question. This angle is often called the reference angle. We often use the Greek letter θ (theta) to represent the reference angle; however, other letters can also be used to represent this angle. The hypotenuse is always the longest side of a right triangle.

A right triangle. The hypotenuse across from the right angle. The opposite side is across from the angle theta. The adjacent side is the third side.


Example 1

Label the hypotenuse, opposite, and adjacent sides of the given triangle.

Table 3.5.1
Given Triangle Solution
A right triangle. The right angle is C, the angle with theta is A, and the third angle is B. The triangle from the original question. Each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.

Each side is labelled with a lowercase letter to match the uppercase letter of the opposite vertex.

c is the hypotenuse

a is the opposite side

b is the adjacent side
Practice 1

Label the sides of each triangle and find the hypotenuse, opposite side, and adjacent side.

a)

A right triangle. The right angle is Y, the angle with theta is Z, and the third angle is X.

 

 

 

 

 

 b)

A right triangle. The right angle is R, the angle with theta is T, and the third angle is S.

 

 

 

 

Trigonometric Ratios

Trigonometric ratios are the ratios of the length of the sides of a triangle. For all right triangles, we can define three trigonometric ratios: sine, cosine, and tangent. The table below outlines the three primary trigonometric ratios.

Table 3.5.2
Primary Trigonometric Ratios
  • sine θ = \frac{\text{the length of the opposite side}}{\text{the length of the hypotenuse side}}
  • cosine θ = \frac{\text{the length of the adjacent side}}{\text{the length of the hypotenuse side}}
  • tangent θ = \frac{\text{the length of the opposite side}}{\text{the length of the adjacent side}}

where θ is the measure of a reference angle measured in degrees

We often use the following abbreviations for the sine, cosine, and tangent ratios:

  • sin θ = \frac{\text{opp}}{\text{hyp}}
  • cos θ = \frac{\text{adj}}{\text{hyp}}
  • tan θ = \frac{\text{opp}}{\text{adj}}

The acronym SOH CAH TOA can be used to help memorize the ratios. As we practice identifying the primary trigonometric ratios, we will label each side with a lowercase letter to match the uppercase letter of the opposite vertex.

Example 2

Find the sine, cosine, and tangent ratios for the given triangle.

Table 3.5.3
Given Triangle Solution
A right triangle. The right angle is D, the angle with theta is F, and the third angle is E. Each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.

sin θ = \frac{\text{f}}{\text{d}}

cos θ = \frac{\text{e}}{\text{d}}

tan θ = \frac{\text{f}}{\text{e}}
Practice 2

For the triangles below, label each side with a lowercase letter to match the uppercase letter of the opposite vertex, then find the sine, cosine, and tangent ratios.

a)

A right triangle. The right angle is Y, the angle with theta is Z, and the third angle is X.

 b)

A right triangle. The right angle is R, the angle with theta is T, and the third angle is S.

When calculating trigonometric ratios, we usually round the ratios to four decimal places unless otherwise stated.

Example 3

For the given triangle, find the sine, cosine, and tangent ratios for angle R and angle S. If necessary, round to four decimal places.

Table 3.5.4
Given Triangle Solution
Right triangle PRS. Right angle at P. Side p=5. Side r=4. Side s=3. Using the definitions, the trigonometric ratios for angle R are

  • sin R= \frac{4}{5} = 0.8
  • cos R= \frac{3}{5} = 0.6
  • tan R=\frac{4}{3} = 1.3333

Using the definitions, the trigonometric ratios for angle S are

  • sin S = \frac{3}{5} = 0.6
  • cos S = \frac{4}{5} = 0.8
  • tan S = \frac{3}{4} = 0.75
Practice 3

For the given triangles, find the sine, cosine, and tangent ratios for both angles in the triangles. If necessary, round to four decimal places.

a)

Right triangle EFD. Right angle at E. Side e=10. Side f=8. Side d=6.

 

 

 b)

Right triangle BAC. Right angle at B. Side b=5.8. Side a=5. Side c=3.

We can also use a calculator to find trigonometric ratios using the sin, cos, and tan buttons. To find the trigonometric ratios, make sure your calculator is in Degree Mode. The steps below show how to put your calculator in Degree Mode.

Table 3.5.5
Degree Mode
  • Press MODE
  • Use the arrows on your calculator to highlight “Degree”
  • Press ENTER
Example 4

Using a calculator, find the trigonometric ratios. If necessary, round to four decimal places.

a) sin 30° b) cos 45° c) tan 60°
Table 3.5.6
Solution
Make sure your calculator is in Degree Mode.

a)  Calculate sin 30° = 0.5

b)  Calculate cos 45° = 0.7071

c)  Calculate tan 60° = 1.7321
Practice 4

Find the trigonometric ratios using your calculator. If necessary, round to four decimal places.

a)  sin 60°

b)  cos 30°

c)  tan 45°

d)  sin 73°

e)  cos 88°

f)  tan 65°

Homework

  1. Label the hypotenuse, opposite side, and adjacent side given the following reference angles.

    a) The reference angle is angle B.

    Right triangle DCB. Angle D is the right angle.

    		 b) The reference angle is angle Z.

    Right triangle YZX. Angle Y is the right angle.

  2. Label the hypotenuse, opposite side, and adjacent side. The reference angle is indicated with θ.

    a)

    Right triangle EFG. Angle E is the right angle. Theta is at G.

    		 b)

    Right triangle EFG. Angle E is the right angle. Theta is at F. Side g is 5.

  3. For the given triangles, find the sine, cosine, and tangent ratios of θ. Use lowercase letters to represent the unknown side lengths.

    a)

    Right triangle EFG. Angle E is the right angle. Theta is at G.

    		 b)

    Right triangle EFG. Angle E is the right angle. Theta is at F. Side g is 5.
    c)

    Right triangle RST. Angle R is the right angle. Theta is at S.

    		 d)

    Right triangle BAC. Angle B is the right angle. Theta is at A.

  4. For the given triangles, find the sine, cosine, and tangent ratios for θ. If necessary, round to four decimal places.

    a)

    A right triangle XZY. X is the right angle. Y is theta. 8 is across from Y. 17 is across from the right angle. 15 is across from Z.

    		 b)

    A right triangle ABC. A is the right angle. B is theta. 2.3 is across from B. 4.7 is across from the right angle. 4.1 is across from C.

  5. For the given triangles, find the sine, cosine, and tangent ratios for both angles. If necessary, round to four decimal places.

    a)

    A right triangle YXZ. Y is the right angle. 4 is across from X. 4.47 is across from the right angle. 2 is across from Z.

     

    		 b)

    A right triangle GFE. G is the right angle. 3.52 is across from E. 6.3 is across from the right angle. 5.23 is across from F.
    c)

    A right triangle YXZ. Y is the right angle. 5 cm is across from X. 7.81 cm is across from the right angle. 6 cm is across from Z.

     

     

    		 d)

    A right triangle POQ. P is the right angle. 4.25 ft is across from O. 7.62 ft is across from the right angle. 6.32 ft is across from Q.

     

  6. Use your calculator to find the given ratios. Round to four decimal places, if necessary.

    a) \sin {47}^{\circ} b) \cos {82}^{\circ}
    c) \tan {12}^{\circ} d) \sin {30}^{\circ}
    e) \tan {53}^{\circ} f) \cos {40}^{\circ}
    g) \sin {81}^{\circ} h) \tan {75}^{\circ}
    i) \cos {36}^{\circ} j) \tan {87}^{\circ}

  7. Which of the three primary trigonometric ratios can have a value larger than one? Explain your answer.

Answers

1. a)
b is the opposite side
c is the adjacent side
d is the hypotenuse		 b)
z is the opposite side
x is the adjacent side
y is the hypotenuse		 2. a)
g is the opposite side
f is the adjacent side
e is the hypotenuse		 b)
f is the opposite side
g is the adjacent side
e is the hypotenuse

3.

a) sin θ  = \frac{g}{e}, cos θ  = \frac{f}{e}, tan θ  = \frac{g}{f} b)  sin θ  = \frac{f}{e}, cos θ  = \frac{g}{e}, tan θ  = \frac{f}{g}
c) sin θ  = \frac{s}{r}, cos θ  = \frac{t}{r}, tan θ  = \frac{s}{t} d) sin θ  = \frac{a}{b}, cos θ  = \frac{c}{b}, tan θ  = \frac{a}{c}

4.

a)
sin θ  = 0.4706
cos θ  = 0.8824
tan θ  = 0.5333		 b)
sin θ  = 0.4894
cos θ  = 0.8723
tan θ  = 0.5610

5.

a)
sin X  = 0.8949
cos X  = 0.4474
tan X  = 2
sin Z  = 0.4474
cos Z  = 0.8949
tan Z  = 0.5		 b)
sin F  = 0.8302
cos F  = 0.5587
tan F  = 1.4858
sin E  = 0.5587
cos E  = 0.8302
tan E  = 0.673		 c)
sin X  = 0.6402
cos X  = 0.7682
tan X  = 0.8333
sin Z  = 0.7682
cos Z  = 0.6402
tan Z  = 1.2		 d)
sin O  = 0.5577
cos O  = 0.8294
tan O  = 0.6725
sin Q  = 0.8294
cos Q  = 0.5577
tan Q  = 1.4871

6.

a) 0.7314 b) 0.1392 c) 0.2126 d) 0.5 e) 1.3270
f) 0.766 g) 0.9877 h) 3.7321 i) 0.809 j) 19.0811
7. The tangent ratio can have a value larger than one if the adjacent side is shorter than the opposite side.

Attribution

Unless otherwise indicated, material on this page has been adapted from the following resources:

Flinn, C., & Overgaard, M. (2020). Math for trades: Volume 2. BCcampus. https://collection.bccampus.ca/textbooks/math-for-trades-volume-2-bccampus-238/, licensed under CC BY 4.0

Mazur, I. (2021). Introductory algebra. BCcampus. https://collection.bccampus.ca/textbooks/intermediate-algebra-bccampus-412/, licensed under CC BY 4.0

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