Lesson 8

Sine and Cosine in the Same Right Triangle

  • Let’s connect cosine and sine.

8.1: Which One Doesn’t Belong: Four Triangles

Which one doesn’t belong?

A

Right triangle P Q R. Q R is x units, P Q is 5 units, angle P Q R is 70 degrees, and angle P R Q is 90 degrees.

B

Right triangle S T U. S U is y units, S T is 6 units, angle S T U is 20 degrees, and angle T S U is 90 degrees.

C

Triangle K L M. K M is x units, L M is 10 units, angle K M L is 115 degrees, and K L M is 15 degrees.

D

Right triangle G H J. G J is x units, G H is 2 point 5 units, and angle G H J is 90 degrees.

8.2: Twin Triangles

Your teacher will assign you to either Column A or Column B. Find the value of the variable for the problems in your column.

Column A:

Column B:

A1

Right triangle. Base is 7 units, hypotenuse is x units. Angle opposite the height is 41 degrees.

B1

Right triangle. Height of 7 units, hypotenuse of x, and angle opposite the height is 49 degrees.

A2

Right triangle. Base is y units, hypotenuse is 6 units. Angle opposite the base is 65 degrees.

B2

Right triangle. Base is y units, hypotenuse is 6 units. Angle opposite the height is 25 degrees.

A3

Right triangle. Height is 8 units, hypotenuse is z units. Angle opposite the height is 50 degrees.

B3

Right triangle. Base is 8 units, hypotenuse is z units. Angle opposite the height is 40 degrees.

Compare your solutions with your group's solutions. Why did you get the same answers to different problems?

8.3: Explain the Co-nnection

  1. Draw a diagram that will help you explain why \(\sin(\theta)=\cos(90 - \theta)\).
  2. Explain why \(\sin(\theta)=\cos(90 - \theta)\).

Discuss your thinking with your group. If you disagree, work to reach an agreement.

Create a visual display that includes:

  • A clearly-labeled diagram.
  • An explanation using precise language.


  1. Make a conjecture about the relationship between \(\tan(\theta)\) and \(\tan(90-\theta)\).
  2. Prove your conjecture.

Summary

In previous lessons, we recalled that any right triangle with acute angles of 25 and 65 degrees was similar to any other right triangle with these same acute angles. Revisiting these triangles, we notice that the sine of 25 degrees is equal to the cosine of 65 degrees, and the cosine of 25 degrees is equal to the sine of 65 degrees.

Right triangle. Base labeled b, height labeled a, hypotenuse labeled c. Angle opposite side a is 25 degrees and angle opposite side b is 65 degrees.
angle cosine of angle = adjacent leg \(\div\) hypotenuse sine of angle = opposite leg \(\div\) hypotenuse
\(25^\circ\) 0.906 0.423
\(65^\circ\) 0.423 0.906

Looking at a general right triangle, the angles can be written as 90, \(\theta\), and \(90-\theta\). Mathematicians often use Greek letters to represent angles. For instance, \(\theta\) is a Greek letter we use frequently in trigonometry.

Right triangle A C B. Right angle at C. A is theta degrees. B is 90 minus theta degrees. A C is x. C B is y. B A is h.
  cosine of angle sine of angle
angle adjacent leg \(\div\) hypotenuse opposite leg \(\div\) hypotenuse
\(\theta^\circ\) \(\frac{x}{h}\) \(\frac{y}{h}\)
\((90-\theta)^\circ\) \(\frac{y}{h}\) \(\frac{x}{h}\)

Glossary Entries

  • cosine

    The cosine of an acute angle in a right triangle is the ratio (quotient) of the length of the adjacent leg to the length of the hypotenuse. In the diagram, \(\cos(x)=\frac{b}{c}\).

  • sine

    The sine of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the hypotenuse. In the diagram, \(\sin(x) = \frac{a}{c}.\)

  • tangent

    The tangent of an acute angle in a right triangle is the ratio (quotient) of the length of the opposite leg to the length of the adjacent leg. In the diagram, \(\tan(x) = \frac{a}{b}.\)

  • trigonometric ratio

    Sine, cosine, and tangent are called trigonometric ratios.