CPM 4.2.3

CPM 4.2.3

How can I determine the angle measure?

Inverse Trigonometry

You now know how to solve for the missing side lengths in a right triangle given an acute angle and the length of any side. But what if you want to determine the measure of an angle? If you are given the lengths of two sides of a right triangle, can you work backwards to determine the measurements of the unknown angles? Today you will work on “undoing” the different trigonometric ratios to determine the angles that correspond to those ratios.

4-78

Mr. Gow needs to build a wheelchair access ramp for the school’s auditorium. The ramp must rise a total of 3 feet to get from the ground to the entrance of the building. In order to follow the state building code, the angle formed by the ramp and the ground cannot exceed 4.76 degrees. 

4-78

Mr. Gow has plans from the planning department that call for the ramp to start 25 feet away from the building.  Will this ramp meet the state building code?

4-78, part a

Draw an appropriate diagram.  Add all the measurements you can.  What does Mr. Gow need to know?

4-78, part b

To determine an angle measure from a trigonometric ratio, you need to “undo” it, just like you can undo addition with subtraction, multiplication with division, or squaring by taking the square root.  These examples are all pairs of inverse operations.

Vocabulary: Inverse Operations

Subtraction is the inverse operation for addition and vice versa, division for multiplication, square root for squaring, and more generally taking the nth root for raising to the nth power.

4-78, part b

When you use a calculator to do this, you use inverse trigonometric functions that are usually labeled “ $\sin^{-1}$  ”, “  $\cos^{-1}$   ”, and “  $\tan^{-1}$   ”.  These are pronounced “inverse sine”, “inverse cosine”, and “inverse tangent”.  On many calculators, you may need to press the “inv” or “2nd” key first, then the “sin”, “cos”, or “tan” key. 

4-78, part b

Verify that your calculator can compute an inverse trig value using the triangle at right. When you compute $\cos^{-1}\left(\frac{8}{16}\right)$ 

 
do you get  $60\degree$ ?

Multiple Choice

Which inverse trig function do we need to use to find the angle for the wheelchair ramp?

• inverse sine  $\left(\sin^{-1}\right)$  

• inverse cosine  $\left(\cos^{-1}\right)$  

• inverse tangent  $\left(\tan^{-1}\right)$  

Multiple Choice

Write an equation to represent the wheelchair problem.

•  $\tan^{-1}\left(4.76\degree\right)=\frac{3}{25}$  
  

•  $\tan^{-1}\left(\frac{3}{25}\right)=4.76\degree$  
  

•  $\tan^{-1}\left(\frac{3}{25}\right)=\theta$  
  

•  $\tan^{-1}\left(\frac{25}{3}\right)=\theta$  

Multiple Choice

In our wheelchair problem, what is the angle of the ramp?

•  $4.76\degree$  
  

• not enough information

•  $6.84\degree$  
  

•  $83.16\degree$  
  

Multiple Choice

The building code says that the angle has to be  $4.76\degree$  or less to be safe. Is the ramp safe?

• Yes, it is safe.

• No, it is not safe.

Multiple Choice

What equation will help us solve for the distance from the building so that the angle of the ramp is  $4.76\degree$  ?

•  $\tan\ 4.76\degree=\frac{x}{3}$  

•  $\tan\ 4.76\degree=\frac{3}{x}$  

•  $\tan^{-1}\left(4.76\degree\right)=\frac{x}{3}$  

•  $\tan^{-1}\left(4.76\right)=\frac{3}{x}$  

4-79

For the triangle at right, determine the measures of ∠A and ∠B. Once you have solved for the measure of the first acute angle (either ∠A or ∠B), what knowledge about the angles in triangles could help you solve for the second acute angle measure?

Multiple Choice

Write an equation and solve for the missing variable.

•  $63\degree$  

•  $27\degree$  

•  $60\degree$  

•  $30\degree$  

Multiple Choice

Write an equation and solve for the missing variable.

•  $90\degree$  

•  $0\degree$  

•  $60\degree$  

•  $30\degree$  

•  $45\degree$  

Multiple Choice

Write an equation and solve for the missing variable.

• 8.35

• 5.87

• 12.85

• 4.55