2/10/17- Inverse Trigonometry

Warm-Up

  1. In the diagrams below, find the values of xy, and z.

 

 

 

 

 

2. a. Identify the reference angle.

b. What words describe the given sides in reference to your angle?

c. What trigonometric function makes use of those sides?

d. Solve for x.

 

 

 

 

Follow-up question from the warm-up:

Find the value of x.

Ask and answer the same three questions as number 2: What is our reference angle? x.

What sides do we have? Opp (6) and adj (10)

What function uses those? Tangent

So we can say tan(x)=6/10, or tan(x)=.6

Now look at your table of values that was passed out in class (http://mrzmath.com/wp-content/uploads/2017/02/trig-table-of-values.pdf)

Think: For #2, we looked up tan(51) on the trig table and got 1.2349.  Now, the question is, what value of x would we plug into the tangent function to get .6 as an output? Look for the output of .6 in the tangent column on your table. We should be able to realize that the value 31º gives us that output, and so conclude that x equals 31º.

Inverse Trigonometry

Inverse trig functions take inputs of ratios of sides and give outputs of angle measures.

The 3 main inverse trig functions are:

arcsin (sin^-1)- The inverse sine function takes an input of the ratio of the side opposite the reference angle to the hypotenuse, and gives the measure of the reference angle.

arccos (cos^-1)- The inverse cosine function takes an input of the ratio of the side adjacent to the reference angle to the hypotenuse, and gives the measure of the reference angle.

arctan (tan^-1)- The inverse tangent function takes an input of the ratio of the side opposite the reference angle to the side adjacent to the reference angle, and gives the measure of the reference angle.

These definitions, along with the example shown below, allowed us to complete the rest of the Blue Trigonometry Practice worksheet passed out the previous day in class.


In summary, we have two ways to think about solving for unknown angle measrues:

  1. Set up the problem as (for example) sin(x)=.7453, and then look at your trig table of values to see what angle measure x as an input to the sine function gives .7453 as an output.
  2. Using the above example, you could instead take the inverse sine of .7453                         [sin^-1(.7453)=x] which would give you the angle measure.

Note: Method 1 can only give you the answer to the nearest degree, while using inverse functions on your calculator can give you a much more accurate answer.

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