5/20/2015
4.3 RIGHT TRIANGLE TRIG.
Students will know how
to use the fundamental
trigonometric identities.
TRIG. FUNCTIONS: A REVIEW
1
5/20/2015
SPECIAL ANGLES FOR SPECIAL
TRIANGLES
As I spoke last class, there are 2 special
triangles: 45° − 45° − 90° and
30° − 60° − 90°. Therefore we have
special scenarios.
𝜋
6
sin 30° = sin =
1
2
Do the rest:
CO-FUNCTIONALITY
Cofunctions of complementary angles are equal. If 𝜃 is an
acute angle, then:
sin(90° − 𝜃) =_________
cos(90° − 𝜃) =_________
tan(90° − 𝜃) =_________
cot(90° − 𝜃) =_________
sec(90° − 𝜃) =_________
csc(90° − 𝜃) =_________
2
5/20/2015
BASIC TRIGONOMETRIC IDENTITIES
The six reciprocal identities:
1
1
sin𝜃 = csc 𝜃
csc 𝜃 = sin 𝜃
cos 𝜃 = sec 𝜃
sec 𝜃 = cos 𝜃
1
1
1
1
tan 𝜃 = cot 𝜃
cot 𝜃 = tan 𝜃
The two quotient identities:
sin 𝜃
tan 𝜃 = cos 𝜃
cot 𝜃 =
cos 𝜃
sin 𝜃
PYTHAGOREAN IDENTITIES
Since 𝑜𝑝𝑝 2 + 𝑎𝑑𝑗
the these identities:
2
= ℎ𝑦𝑝
2
We can divide both sides by ℎ𝑦𝑝
2
to
sin2 𝜃 + cos 2 𝜃 = 1,
tan2 𝜃 + 1 = sec 2 𝜃,
1 + c𝑜𝑡 2 𝜃 = c𝑠𝑐 2 𝜃,
As mentioned before:
An angle of elevation is . . . the angle from the horizontal upward to an object.
An angle of depression is . . . the angle from the horizontal downward to an object.
3
5/20/2015
EXAMPLES: APPLYING IDENTITIES
Left 𝜃 be an acute angle such that sin 𝜃 = 0.7. Find the values of:
A) cos 𝜃
B) tan 𝜃
USE IDENTITIES:
Use trigonometric identities to transform one side of the equation into the
other.
A) s𝑒𝑐 𝜃 cos 𝜃 = 1
B) (sec 𝜃 + tan 𝜃) sec 𝜃 − tan 𝜃 = 1
4
5/20/2015
USING CALCULATORS
Evaluate: sec(5°40′ 12′′ )
4.4 TRIG FUNCTIONS AT ANY
ANGLE
Students will know how to
evaluate trigonometric
functions for any angle.
5
5/20/2015
TRIG FUNCTIONS
Let θ be angle in standard position with (x, y) a point on the terminal side
of θ and 𝑟 =
𝑥 2 + 𝑦2 ≠ 0 .
sin 𝜃 = ________
cos θ = __________
tan 𝜃 =_______
cot 𝜃 = __________
sec 𝜃 =_______
csc 𝜃 =_______
EVALUATING TRIGONOMETRIC
FUNCTIONS
Let (−3,4) be equal to (𝑥, 𝑦) on the point on the terminal side
of 𝜃. Find Sine, Cosine, and tangent of 𝜃.
6
5/20/2015
QUADRANTS
Name the quadrants in which the sine function
is positive.
Name the quadrants in which the sine function
is negative.
Name the quadrants in which the cosine
function is positive.
Name the quadrants in which the cosine
function is negative.
Name the quadrants in which the tangent
function is positive.
Name the quadrants in which the tangent
function is negative.
EVALUATING TRIGONOMETRIC
FUNCTIONS
5
Given that tan 𝜃 = − and cos 𝜃 > 0,
4
find sin 𝜃 and sec 𝜃. Use the exact
values
Note that 𝜃 lies in quadrant IV
because that is the only quadrant
in which the tan is negative and
the cosine is positive.
7
5/20/2015
THE TRIGONOMETRIC QUADRANTS
Evaluate the sine and cosine functions at
𝜋
3𝜋
the four quadrant angles 0, , 𝜋,
2
2
REFERENCE ANGLES
If 𝜃 is in standard position, then the reference angle 𝜃′ associated with 𝜃 is
the acute angle formed by the terminal side of 𝜃 and the x-axis.
8
5/20/2015
EXAMPLE
Example a. Find the reference angle for the following angles.
a) 𝜃 = 125°
Example b. Find the reference angle for the following angles.
a) 𝜃 = 5
TRIG. OF ANY ANGLE
To find the value of a trigonometric function of any angle 𝜃, . . .
determine the function value for the associated reference angle
𝜃′.Depending on the quadrant in which 𝜃 lies, affix the appropriate sign to
the function value.
IF 𝜃 is in standard position with (x, y) on the terminal, then if 𝜃′ is
placed in standard position (|x|, |y|) will be on its terminal side. Also, the r
for 𝜃 and the r for 𝜃′ will be the same. Hence, |sin 𝜃 | = sin 𝜃′ , |cos 𝜃 | =
cos 𝜃 ′ ,|tan 𝜃 | = tan 𝜃′.
This means that to evaluate trigonometric functions for any angle, we
need only find the value of that function for the reference angle and
attach the proper sign according to the quadrant in which 𝜃 lies.
9
5/20/2015
EXAMPLES
Use pythag. Identities
Example 5. Let q be an angle in the third
quadrant so that cos𝜃 = -1/4.
Find the following.
a) sin𝜃
HOMEWORK
p.310-312 #1-9odd, 19, 20, 25-33,
37,43,47-51odd, 57,61
p.320#1,5,7
10