Lesson 9

Using Trigonometric Ratios to Find Angles

Lesson Narrative

For the past few lessons students have been finding side lengths in right triangles where they are given an acute angle and the length of one other side. In this lesson they find an angle given two side lengths of a right triangle. They learn to use a calculator to look up the angle corresponding to a ratio of sides and then apply that skill to a variety of problems and a context.

As students grapple with the idea of using sides to find angles, they will need to move from concrete to abstract. The arctangent, arccosine, or arcsine of a ratio tells them an angle measure in any right triangle with that particular ratio of side lengths. Students continue to reason abstractly and quantitatively (MP2) as they apply these ideas to the ladder scenario in the next activity, especially if they do the measurements themselves.

Some calculators and texts use arctan and others use \(\tan^{\text-1}\). The lesson summary uses the arc notation because it is clearer. Decide whether to introduce one or both notations in the synthesis of the warm up. In these materials \(arccos(\theta)\), \(arcsin(\theta)\), and \(arctan(\theta)\) are used so students don’t confuse \(x^{\text-1}\) which means the reciprocal of \(x\) with \(\tan^{\text-1}(\theta)\) which does not mean the reciprocal of tangent. If your calculators (or an exam your students will encounter) use \(\sin^{\text-1}(\theta), \cos^{\text-1}(\theta), \) and \(\tan^{\text-1}(\theta)\) then use that notation, but explain the potential for confusion and clarify the difference between the notations.

This is the first lesson in which students encounter angle measurements produced by calculators using arccosine, arcsine, and arctangent. The calculator will provide an unreasonable number of decimal places in the output for most input ratios, and students have an opportunity to reason about how many decimal places to report. Because we have been in the habit of measuring angles using protractors which are precise to the nearest degree, we adopt the convention of rounding to the nearest whole number.

Learning Goals

Teacher Facing

  • Calculate angle measures in right triangles using arccosine, arcsine, and arctangent.
  • Describe how to apply the safety guidelines for ladders to solve problems (using words and other representations).

Student Facing

  • Let’s work backwards to find angles in right triangles.

Learning Targets

Student Facing

  • I can use arccosine, arcsine, and arctangent to find angle measures in right triangles.

CCSS Standards

Building On

Addressing

Glossary Entries

  • arccosine

    The arccosine of a number between 0 and 1 is the acute angle whose cosine is that number.

  • arcsine

    The arcsine of a number between 0 and 1 is the acute angle whose sine is that number.

  • arctangent

    The arctangent of a positive number is the acute angle whose tangent is that number.

  • 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.