Transcript Example 1

Section 5-5: Solving Right Triangles
Your calculator can find sine, cosine, and tangent in degrees and radians.
sin
cos
tan
We will use their definition to find secant, cosecant, and cotangent.
csc =
1
𝑠𝑖𝑛
1
sec =
𝑐𝑜𝑠
cot =
1
𝑡𝑎𝑛
When using the calculator, first you must determine if the angle is in DEGREES or RADIANS. Put your calculator
in the appropriate MODE.
Example 1:
a)
Use the calculator to find the following.
cot 45°
1
= 1
tan 45°
b)
csc 30°
1
= 2
sin 30°
INVERSE OF THE SINE, COSINE AND TANGENT FUNCTIONS
How do we solve :
sin 𝜃 = -.5
𝜃 = sin−1(−.5)
𝜃 = -30°
c)
sec 22.5°
1
=
cos 22.5°
1.0823922
That means, sin-1 and sin will cancel each other out, as well as cos and cos-1, or tan and tan-1.
We go backward on the calculator to find the angle in degrees or radians using the
sin-1
called arcsine
cos-1
tan-1
called arccosine
called arctangent
relations
Arcsine, arccosine and arctangent are ____________not functions.
When going backward, the best method for understanding your key sequence on the
calculator is to solve algebraically for 𝜽 first, then type into the calculator.
Example 2:
Find in degrees.
a) cos 𝜃 = .5
𝜃 = cos −1 (.5)
𝜃 = 60°
b)
sin 𝜃 = .7216
c)
tan 𝜃 = 1.1256
𝜃 = sin−1 (.7216)
𝜃 = tan−1 (1.1256)
𝜃 = 46.18673788
𝜃 = 48.38162965
𝜃 ≈ 46.2°
𝜃 ≈ 48.4°
Example 1:
If r = 14 and s = 8, find S.
You want to find the measure of an acute angle in a
right triangle. You know the side opposite the angle
and the hypotenuse.
sin S =
𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
sin S =
8
14
S = 𝑠𝑖𝑛−1
4
7
S = 34.8499
Angle S is about 34.8°.
Trigonometry can be used to find the angle of ______________ or the angle of
depression.
Example 2:
HOUSEHOLD The camera for a baby monitor is set up on a shelf in a child’s room, and
it is angled so that it captures the image of the center of the baby’s crib.
The shelf is about 3 feet higher than the crib, and its horizontal distance
from the crib is about 7 feet. What is the angle of depression of the light?
The angle of depression and the angle of elevation
are equal in measure because they are alternate
interior angles.
𝑠𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
tan 𝜃 =
𝑠𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
tan 𝜃 =
3
7
𝜃 = 𝑡𝑎𝑛−1
3
7
𝜃 = 23.1986
The angle of depression should be about 23.2°.
You can use trigonometric functions and inverse relations to solve right triangles. To solve a
right triangle means to find __________________________
all the measures of its sides and ____________.
the angles
*** Whenever possible, use the measures given in the problem to find the unknown measures.
Example 3:
Solve each triangle described, given the triangle at the right.
A.
Find
B = 42°, b = 4.5

Find a :
A :

A + 42 + 90= 180

A = 48°
tan 48° =
𝑎
4.5
4.5 tan 48° = 𝑎
a = 4.9978
𝑎 ≈5
Find c : cos 48° =
4.5
𝑐
𝑐 cos 48° = 4.5
𝑐=
4.5
cos 48°
c = 6.7251
𝑐 ≈ 6.7
B.
b = 18, c = 52
Find a :
Find A :
a2 + (18)2 = (52)2
a2 + 324 = 2704
a2 = 2380
a = 2 595
𝑎 ≈ 48.7852
𝑎 ≈ 48.8
sin 𝐴 =
48.8
52
48.8
A = 𝑠𝑖𝑛−1 52
A = 69.7948
A = 69.8°
Find B : 69.8° + 𝐵 + 90° = 180°
B = 20.2°