Precalculus by Richard Wright

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4-03 Right Triangle Trigonometry

Mr. Wright teaches the lesson.

Summary: In this section, you will:

SDA NAD Content Standards (2018): PC.5.3

Navy men work on school in Ecuador.
Navy men on step ladders working on a school in Ecuador. (United States Navy)

A 8-foot step ladder actually is not 8-feet high. The size of a step ladder is actually the length of the rails the steps are attached to. When the ladder is in use, the rails are slanted so the height is less. Using right triangles we can calculate the actual height of an 8-foot step ladder.

Right Triangle Trigonometry

A right triangle has one right angle and two acute angles. The side opposite the right angle is the hypotenuse and is the longest side. The other sides are called legs. If one of the acute angles is chosen the leg forming one side of the angle is called the adjacent leg. The leg opposite from the chosen angle is called the opposite leg.

Right Triangle
Parts of a right triangle

The six trigonometric functions have definitions similar to the unit circle, only this time the angle must be acute and the fraction part of the formulas is a ratio of sides of the triangle.

Trigonometric Functions of a Right Triangle
sin θ = opp hyp csc θ = hyp opp
cos θ = adj hyp sec θ = hyp adj
tan θ = opp adj cot θ = adj opp

Where opp = length of the opposite leg, adj = length of the adjacent leg, and hyp = length of the hypotenuse.

Notice that sine and cosecant, cosine and secant, tangent and cotangent are reciprocals just like they were with the unit circle.

Evaluate Trigonometric Functions

Evaluate the six trigonometric functions of α for the triangle in figure 3.

Solution

Use the Pythagorean Theorem to solve for the third side.

adj 2 + opp 2 = hyp 2 8 2 + opp 2 = 10 2 opp = 10 2 - 8 2 opp = 36 opp = 6

From the figure, adj = 8, hyp = 10, and opp = 6. Use the formulas to evaluate the trigonometric functions.

sin α = opp hyp = 6 10 = 3 5 csc α = opp hyp = 10 6 = 5 3
cos α = adj hyp = 8 10 = 4 5 sec α = opp hyp = 10 8 = 5 4
tan α = opp adj = 6 8 = 3 4 cot α = adj opp = 8 6 = 4 3

Evaluate the six trigonometric functions of β for the triangle in figure 4.

Answers

sin β = 2 13 13 csc β = 13 2
cos β = 3 13 13 sec β = 13 3
tan β = 2 3 cot α = 3 2

Special Right Triangles

Two special triangles contain the most common angles of 30°, 45°, and 60°. The special triangles can be used to evaluate the trigonometric functions of those angles.

45°-45°-90° Triangle

Evaluate the six trigonometric functions of 45°.

Solution

Draw a right triangle with an acute angle of 45°. The acute angles of a right triangle are complementary, so the other acute angle is 45°. Thus this is an isosceles triangle and both legs are the same length. Use the Pythagorean Theorem to find the length of the hypotenuse.

hyp 2 = 1 2 + 1 2 hyp = 2

45°-45°-90° triangle
sin 45 ° = opp hyp = 1 2 = 2 2 csc 45 ° = opp hyp = 2 1 = 2
cos 45 ° = adj hyp = 1 2 = 2 2 sec 45 ° = opp hyp = 2 1 = 2
tan 45 ° = opp adj = 1 1 = 1 cot 45 ° = adj opp = 1 1 = 1

30°-60°-90° Triangle

Evaluate the six trigonometric functions of 30°.

Solution

Draw an equilateral triangle with an angles of 60°. Draw a vertical line to bisect the triangle and create a 30°-60°-90° triangle. Set each side of the equilateral triangle as 2, so the short leg of the 30°-60°-90° triangle is 1. Use the Pythagorean Theorem to find the length of the longer leg.

x 2 + 1 2 = 2 2 x = 2 2 - 1 2 x = 3

30°-60°-90° triangle
sin 30 ° = opp hyp = 1 2 csc 30 ° = opp hyp = 2 1 = 2
cos 30 ° = adj hyp = 3 2 sec 30 ° = opp hyp = 2 3 = 2 3 3
tan 30 ° = opp adj = 1 3 = 3 3 cot 30 ° = adj opp = 3 1 = 3

Evaluate the six trigonometric functions of 60°.

Answers

sin 60 ° = opp hyp = 3 2 csc 30 ° = opp hyp = 2 3 3
cos 30 ° = adj hyp = 1 2 sec 30 ° = opp hyp = 2 3 = 2 1 = 2
tan 30 ° = opp adj = 1 3 = 3 cot 30 ° = adj opp = 3 1 = 3 3
Sine, Cosine, and Tangent of Special Angles
sin 30 ° = sin π 6 = 1 2 cos 30 ° = cos π 6 = 3 2 tan 30 ° = tan π 6 = 3 3
sin 45 ° = sin π 4 = 2 2 cos 45 ° = cos π 4 = 2 2 tan 45 ° = tan π 4 = 1
sin 60 ° = sin π 3 = 3 2 cos 60 ° = cos π 3 = 1 2 tan 60 ° = tan π 3 = 3
Lesson Summary

Trigonometric Functions of a Right Triangle
sin θ = opp hyp csc θ = hyp opp
cos θ = adj hyp sec θ = hyp adj
tan θ = opp adj cot θ = adj opp

Where opp = length of the opposite leg, adj = length of the adjacent leg, and hyp = length of the hypotenuse.


Sine, Cosine, and Tangent of Special Angles
sin 30 ° = sin π 6 = 1 2 cos 30 ° = cos π 6 = 3 2 tan 30 ° = tan π 6 = 3 3
sin 45 ° = sin π 4 = 2 2 cos 45 ° = cos π 4 = 2 2 tan 45 ° = tan π 4 = 1
sin 60 ° = sin π 3 = 3 2 cos 60 ° = cos π 3 = 1 2 tan 60 ° = tan π 3 = 3

Helpful videos about this lesson.

Practice Exercises

  1. Draw a right triangle and label one acute angle θ. Label the adjacent, opposite, and hypotenuse.
  2. Evaluate the six trigonometric functions for the indicated angles.
  3. Use the special right triangles to evaluate the indicated trigonometric function.
  4. sin 30°
  5. csc 45°
  6. cot 60°
  7. sec 30°
  8. cosπ4
  9. secπ3
  10. cotπ4
  11. cscπ6
  12. Mixed Review
  13. (4-02) Using the unit circle, evaluate sec3π2.
  14. (4-02) Using the unit circle, evaluate sin 570°.
  15. (4-01) Draw the angle, 7π4 in standard position, then find a positive and negative coterminal angle.
  16. (3-04) Solve log(x) − log(x + 2) = 1.
  17. (2-01) Divide 2-ii.

Answers

  1. sin α = 5 13 , cos α = 12 13 , tan α = 5 12 , csc α = 13 5 , sec α = 13 12 , cot α = 12 5
  2. sin α = 4 5 , cos α = 3 5 , tan α = 4 3 , csc α = 5 4 , sec α = 5 3 , cot α = 3 4
  3. sin β = 15 17 , cos β = 8 17 , tan β = 15 8 , csc β = 17 15 , sec β = 17 8 , cot β = 8 15
  4. sin β = 6 61 61 , cos β = 5 61 61 , tan β = 6 5 , csc β = 61 6 , sec β = 61 5 , cot β = 5 6
  5. sin α = 5 41 41 , cos α = 4 41 41 , tan α = 5 4 , csc α = 41 5 , sec α = 41 4 , cot α = 4 5
  6. sin β = 2 22 13 , cos β = 9 13 , tan β = 2 22 9 , csc β = 13 22 44 , sec β = 13 9 , cot β = 9 22 44
  7. 1⁄2
  8. 2
  9. 33
  10. 233
  11. 22
  12. 2
  13. 1
  14. 2
  15. undefined
  16. -12
  17. ; 15π4; -π4
  18. No solution
  19. −1 − 2i