Download Lesson 6-2 Solving Side Lengths with Trig

Math 2 Honors
Name ________________________
Lesson 6-2: Solving Side Lengths of Triangles with Trigonometric Functions
Learning Goals:


I can calculate sine and cosine ratios for acute angles in a right triangle when given two side
lengths.
I can use sine, cosine, tangent to solve for the unknown side lengths of a right triangle.
1. The following sketch shows the start of one surveyor’s attempt to determine the height of a tall
mountain without climbing to the top herself.
a. Use the given information to calculate the lengths of AB and BC. Use the table from Problem 8
in Lesson 5-1 to calculate approximate values of trigonometric functions as needed.
AB =
BC =
b. Suppose that a laser ranging device allowed you to find the length of AB and the angle of
elevation BAC, but you could not measure the length of AC. How could you use this
information (instead of the information from the diagram) to calculate the lengths of AC and
BC ? Solve for AC and BC :
2. The trigonometric functions are often used in problems modeled with right triangles, as in
Problem 1. It is helpful to be able to use these functions without first placing an acute angle of the
triangle in standard position in a coordinate plane. Examine the diagram of right ABC with C a
right angle.
a. Explain why the following right triangle definitions of sine, cosine, and tangent make sense.
tangent of A  tan A 
sine of A  sin A 
a
length of side opposite A

b
length of side adjacent to A
a
length of side opposite A

c
length of hypotenuse
cosine of A  cos A 
b
length of side adjacent A

c
length of hypotenuse
b. Write expressions for tan B, sin B, and cos B.
tan ∠B =
sin ∠ B =
cos ∠ B =
3. Chicago’s Bat Column, a sculpture by Claes Oldenburg, is shown below.
a. About how tall do you think the column is? What visual clues in the photo did you use to make
your estimate?
b. In the diagram at the right, what lengths and angles could you determine easily by direct
measurement (and without using high-powered equipment)?
c. Which trigonometric functions of A involve side BC ? Of these, which also involve a
measurable length?
d. Which of the trigonometric functions of B involve side BC and a measurable length? If you
know the measure of angle of elevation A, how can you find the measure of B ?
e. To find the height of Bat Column, Krista and D’wan proceeded as follows. First, Krista chose a
spot (point A) 20 meters from the sculpture (point C). D’wan estimated the angle of elevation at
A by sighting the top of the sculpture along a protractor and using a weight as shown. He
measured A to be 55. What is the measure of B ?
f. They next used the following reasoning to find the height BC.
We need to find BC. We know that tan 55 
BC
.
AC
But, tan 55  1.4281 and AC  20 m.
BC
.
So, we need to solve 1.4281 
20
If we multiply both sides of the equation by 20,
we get BC  1.4281 20, or about 29 m.
i.
Why did they decide to use the tangent function rather than the sine function?
ii.
How did they know that tan 55  1.4281?
iii.
Check that each step in their reasoning is correct.
iv.
How do you think Krista and D’wan used this information to calculate the height of Bat
Column?
g. Ken said he could find the length AB (the line of sight distance) by solving cos 55 
AC
.
AB
i.
Use Ken’s idea to find the length AB.
ii.
What is another way you could find AB by using a different trigonometric function?
iii.
Could you find AB without using a trigonometric function? Explain your reasoning.
4. As you have seen in Problems 1 and 3, an important part of solving problems using trigonometric
methods is to decide on a trigonometric function that uses given information. For each right triangle
below, write two equations involving trigonometric functions of acute angles that include s and the
indicated length. Then rewrite each equation in an equivalent form “s = …”.
a.
b.
c.
5. Rather than using tables, today it is much easier to find values of the trigonometric functions with a
calculator. To calculate a trigonometric function value for an angle measured in degrees, first be
sure your calculator is set in degree mode.
Press: c  5: Settings  2: Document Settings…  Choose Degree  Make Default  OK
a. Calculate the following using the calculator. The trigonometric functions can be found by
pressing the µ key.
i.
sin 27.5o =
ii.
cos 27.5o =
iii. tan 27.5o =
b. Use your calculator to find the following trig functions and compare your results with the table
found in Lesson 5-1 Problem 8.
sin 66 o =
sin 54 o =
cos 66 o =
cos 54 o =
tan 66 o =
tan 54 o =
c. Use your calculator to find the following trig functions and compare you results with the exact
values you found in Lesson 5-1 Problem 6.
sin 45 o =
cos 45 o =
tan 45 o =
6. Each part below gives angle measure and side length information for right ABC with C a right
angle. For each, label the triangle and then find the lengths of the remaining two sides and find the
measure of the third angle.
a. B  52, a  5 m
b. A  48, a  15 mi
c.
A  31, b  8 in
d. A  70, c  14 cm