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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 =
π‘π‘œπ‘ 
cot =
1
π‘‘π‘Žπ‘›
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 πœƒ = -.5
πœƒ = sinβˆ’1(βˆ’.5)
πœƒ = -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 πœƒ = .5
πœƒ = cos βˆ’1 (.5)
πœƒ = 60°
b)
sin πœƒ = .7216
c)
tan πœƒ = 1.1256
πœƒ = sinβˆ’1 (.7216)
πœƒ = tanβˆ’1 (1.1256)
πœƒ = 46.18673788
πœƒ = 48.38162965
πœƒ β‰ˆ 46.2°
πœƒ β‰ˆ 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 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’
β„Žπ‘¦π‘π‘œπ‘‘π‘’π‘›π‘’π‘ π‘’
sin S =
8
14
S = π‘ π‘–π‘›βˆ’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 π‘œπ‘π‘π‘œπ‘ π‘–π‘‘π‘’
tan πœƒ =
𝑠𝑖𝑑𝑒 π‘Žπ‘‘π‘—π‘Žπ‘π‘’π‘›π‘‘
tan πœƒ =
3
7
πœƒ = π‘‘π‘Žπ‘›βˆ’1
3
7
πœƒ = 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° =
π‘Ž
4.5
4.5 tan 48° = π‘Ž
a = 4.9978
π‘Ž β‰ˆ5
Find c : cos 48° =
4.5
𝑐
𝑐 cos 48° = 4.5
𝑐=
4.5
cos 48°
c = 6.7251
𝑐 β‰ˆ 6.7
B.
b = 18, c = 52
Find a :
Find A :
a2 + (18)2 = (52)2
a2 + 324 = 2704
a2 = 2380
a = 2 595
π‘Ž β‰ˆ 48.7852
π‘Ž β‰ˆ 48.8
sin 𝐴 =
48.8
52
48.8
A = π‘ π‘–π‘›βˆ’1 52
A = 69.7948
A = 69.8°
Find B : 69.8° + 𝐡 + 90° = 180°
B = 20.2°