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?
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:
Compare your solutions with your group's solutions. Why did you get the same answers to different problems?
8.3: Explain the Connection
 Draw a diagram that will help you explain why \(\sin(\theta)=\cos(90  \theta)\).
 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 clearlylabeled diagram.
 An explanation using precise language.
 Make a conjecture about the relationship between \(\tan(\theta)\) and \(\tan(90\theta)\).
 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.
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.
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.