Lesson 6

Working with Trigonometric Ratios

  • Let’s solve problems using cosine, sine, and tangent.

6.1: This Time with Strategies

Estimate the value of \(z\).

Right triangle GHJ. Angle G = 90 degrees, angle J= 40 degrees, angle H= 50 degrees. Side GH=3 units. Side HJ labeled z.

6.2: New Names, Same Ratios

  1. Use your calculator to determine the values of \(\cos(50)\)\(\sin(50)\), and \(\tan(50)\).
  2. Use your calculator to determine the values of \(\cos(40)\)\(\sin(40)\), and \(\tan(40)\).
  3. How do these values compare to your chart? 
  4. Find the value of \(z\).
Right triangle GHJ. Angle G = 90 degrees, angle J= 40 degrees, angle H= 50 degrees. Side GH=3 units. Side HJ labeled z.

6.3: Solve These Triangles

  1. Solve for \(x\).

    Right triangle ABC. Angle A= 90 degrees, angle B=17 degrees. Side AB = 10 units. Side BC labeled x.
  2. Solve for \(y\).

    triangle DEF. Angle D = 42 degrees, angle E = 48 degrees. Side DE=7 units. Side DF labeled y.
  3. Find all the missing sides and angle measures.

    1. The measure of angle \(X\) is 90 degrees and angle \(Y\) is 12 degrees. Side \(XZ\) has length 2 cm.

    2. Right triangle N P Q. N Q is 4 units, angle N Q P is 90 degrees, angle P N Q is 60 degrees.
    3. The measure of angle \(K\) is 90 degrees and angle \(L\) is 71 degrees. Side \(LM\) has length 20 cm.



Complete the table.

angle cosine sine tangent
\(80^\circ\)      
\(85^\circ\)      
\(89^\circ\)      

Based on this information, what do you think are the cosine, sine, and tangent of 90 degrees? Explain or show your reasoning.

Summary

We have a column in the right triangle table for "adjacent leg \(\div\) hypotenuse." We use this ratio so frequently it has a name. It is called the cosine of the angle. We write \(\cos(25)\) to say the cosine of 25 degrees. A scientific calculator can display the cosine of any angle. This means we can more precisely calculate unknown side lengths rather than estimating using the table. The right triangle table is sometimes called a trigonometry table since cosine, sine, and tangent are trigonometric ratios. Here is what the table looks like with the ratios labeled with their special names:

  cosine sine tangent
angle adjacent leg \(\div\) hypotenuse opposite leg \(\div\) hypotenuse opposite leg \(\div\) adjacent leg
\(25^\circ\) \(\cos(25)=0.906\) \(\sin(25)=0.423\) \(\tan(25)=0.466\)
Right triangle. Sides labeled a,b,c. Angle opposite side c= 90 degrees, angle opposite side a = 25 degrees.

If the length \(b\) is 7, we can find \(c\) by solving the equation \(\cos(25)=\frac{7}{c}\). So \(c\) is about 7.7 units. To solve for \(a\) we have 3 choices: the Pythagorean Theorem, sine, and tangent. Let’s use tangent by solving the equation \(\tan(25)=\frac{a}{7}\). So \(a\) is about 3.3 units. We can check our answers using the Pythagorean Theorem. It should be true that \(3.3^2+7^2=7.7^2\). The two expressions are almost equal, which makes sense because we expect some error due to rounding.

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.