Lesson 2: The Sine and Cosine Ratios - Solving Unknown Angle Measures Using Sine or Cosine

   Constructing Knowledge

Recall the steps used to solve for the angle measure when using the tangent ratio from Lesson 1.

Step 1: Identify and label the sides as being adjacent to, opposite or the hypotenuse in relation to the angle indicated.

Step 2: State the appropriate ratio.

Step 3: Substitute known values and calculate the unknown value.

   Multimedia

A video using trigonometry to solve an unknown angle is provided.



EXAMPLE 1

Determine the measure of angle θ, to the nearest degree.



Solution


Step 1: Identify and label the sides as being adjacent to, opposite or the hypotenuse in relation to the angle indicated.



Step 2: State the appropriate ratio.

The length opposite θ and the hypotenuse are involved, so use the sine ratio.

\(\text{sin}\,\theta=\frac{\text{length opposite}\,\theta}{\text{hypotenuse}}\)

Step 3: Substitute known values and calculate the unknown value.

\(\begin{align} \text{sin}\,\theta&=\frac{\text{length opposite}\,\theta}{\text{hypotenuse}} \\ \\ \text{sin}\,\theta&=\frac{11}{17} \\ \\ \theta&=\text{sin}^{-1}\left(\frac{11}{17}\right) \\ \\ \theta&=40.32° \\ \\ \theta&=40° \\ \end{align}\)

The measure of angle θ is approximately 40°.

EXAMPLE 2

Determine the measure of angle θ, to the nearest degree.



Solution


Step 1: Identify and label the sides as being adjacent to, opposite or the hypotenuse in relation to the angle indicated.



Step 2: State the appropriate ratio.

The length adjacent to θ and the hypotenuse are involved, so use the cosine ratio.

\(\text{sin}\,\theta=\frac{\text{length opposite}\,\theta}{\text{hypotenuse}}\)

Step 3: Substitute known values and calculate for the unknown value.

\(\begin{align} \text{cos}\,\theta&=\frac{\text{length adjacent to}\,\theta}{\text{hypotenuse}} \\ \\ \text{cos}\,\theta&=\frac{5.5}{11.0} \\ \\ \theta&=\text{cos}^{-1}\left(\frac{5.5}{11.0}\right) \\ \\ \theta&=60° \\ \end{align}\)

The measure of angle θ is 60°.


Now, it is your turn! Complete the questions in your Chapter 4, Lesson 2 Practice Makes Perfect that refer to Solving Unknown Angle Measures Using Sine or Cosine.



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