Right Triangle Trigonometry
Algebra III, Sec. 4.3
Objective
Evaluate trigonometric functions of acute angles;
Use the fundamental trigonometric identities.
opposite
The Six Trigonometric Functions
opposite
sin q =
hypotenuse
hypotenuse
cscq =
opposite
cosq =
adjacent
hypotenuse
secq =
hypotenuse
adjacent
tanq =
opposite
adjacent
cot q =
adjacent
opposite
θ
adjacent
The Six Trigonometric Functions
The cosecant (csc) function is the reciprocal of the
sine
______________
function.
The cotangent (cot) function is the reciprocal of the
tangent function.
_____________
The secant (sec) function is the reciprocal of the
cosine
_____________
function.
Example 1
Find the values of the six trigonometric functions of θ.
5
2
θ
First, find the missing side… a2 + b2 = c 2
a2 + 22 = 52
Then, find the six trig fns… sinq =
cosq =
tanq =
a2 = 21
cscq =
secq =
cot q =
a = 21
Example 2
Find the values of cot 45° and csc 45°.
1, √2
Example 3
Use the equilateral triangle shown to find the values
of cot 60° and cot 30°.
30°
2
√3
60°
1
√3/3, √3
Example (on your handout)
In the right triangle below, find sinθ, cosθ, and tanθ.
sinq =
cosq =
12
θ
5
tanq =
Cofunctions
equal
Cofunctions of complementary angles are _______.
sin ( 90° - q ) = cosq
cos ( 90° - q ) = sinq
tan ( 90° - q ) = cot q
cot ( 90°- q ) = tanq
sec ( 90° - q ) = cscq
csc ( 90° - q ) = secq
Example 4
Use a calculator to evaluate…
cot 34° 30’ 26”.
1.4545
Reciprocal Identities
1
sin q =
cscq
cscq =
1
sinq
1
secq
secq =
1
cosq
cosq =
1
tanq =
cot q
1
cot q =
tan q
Quotient Identities
sin q
tanq =
cosq
cosq
cot q =
sinq
Pythagorean Identities
sin 2 q + cos2 q =1
1+ tan q = sec q
2
2
1+ cot q = csc q
2
2
Example 5
Let θ be an acute angle such that cosθ = 0.96.
Find the values of (a) sinθ and (b) tanθ, using
trigonometric identities.
(a) 0.28
(b) 0.2916
Example 6
Let β be an acute angle such that tanβ= 4. Find
the values of (a) cotβ and (b) secβ, using
trigonometric identities.
(a) ¼
(b) √17
Example 7
Use trigonometric identities to transform the left side
of the equation into the right side (0 < θ < π/2).
a.
secq tanq
=1
cosq cot q
Example 7
Use trigonometric identities to transform the left side
of the equation into the right side (0 < θ < π/2).
b.
(sinq + cosq ) + (sinq - cosq )
2
2
=2
Applications
 What does it mean to solve a right triangle?
 Find all of the sides and angles!
 The term angle of elevation means…
the angle from the horizontal upward to an object.
 The term angle of depression means…
the angle from the horizontal downward to an object.
Applications: EXAMPLE 1
 Solve ΔXYZ, given
X
z = 20
y
Z
x
Y
ANSWER 1
X
z = 20
y
Z
x
Y
Complementary angles
Applications: EXAMPLE 2
 Solve ΔXYZ, given
X
y = 45
Z
z
x = 28
Y
 Hint: Use Calculator to
change to a decimal.
Then use inverse key.
You always use the
inverse key to find angle.
X
y = 45
Z
z
x = 28
Y
Applications: EXAMPLE 3
A surveyor found that the angle of elevation of the
top of a flagpole was
. The observation was
made from a point 1.5 m above ground and 10 m
from the base of the flagpole. Find the height of the
flagpole to the nearest tenth of a meter.
ANSWER 3
Applications: EXAMPLE 4
The angle of depression from the top of a cliff 800 m
high to the base of a log cabin is
. How far is the
cabin from the foot of the cliff?
Alternate interior angles are congruent
Practice!
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