Pre-5.1 - trigonometry ratios in right triangle and special right triangles.ppt

TRIGONOMETRIC RATIOS

1. Right Triangle Trigonometry
Trigonometry is based upon ratios of the sides
of right triangles.
The six trigonometric functions of a right triangle,
with an acute angle ,are defined by ratiosof two
sides of the triangle.
θ
opp
hyp
adj
The sides of the right triangle are:
the side opposite the acute angle
,
the side adjacentto the acute angle ,
and the hypotenuseof the right
triangle.

2. A
A
The hypotenuse is the longest side and is always
opposite the right angle.
The opposite and adjacent sides refer to another
angle, other than the 90
o
.
Right Triangle Trigonometry

3. S O HC A HT O A
The trigonometric functions are:
sine, cosine, tangent, cotangent, secant, and
cosecant.
opp
adj
hyp
θ
sin = cos = tan =
csc = sec = cot =
opp
hyp
adj
hyp
hyp
adj
adj
opp
opp
adjhyp
opp
Trigonometric Ratios

4. Finding an angle from a triangle
To find a missing angle from a right-angled triangle
we need to know two of the sides of the triangle.
We can then choose the appropriate ratio, sin, cos
or tan and use the calculator to identify the angle
from the decimal value of the ratio.
Find angle C
a)Identify/label the names
of the sides.
b) Choose the ratio that
contains BOTH of the
letters.
14 cm
6 cm
C
1.

5. C = cos
-1
(0.4286)
C = 64.6
o
14 cm
6 cm
C
1.
h
a
We have been given the
adjacent and hypotenuse
so we use COSINE:
Cos A = hypotenuse
adjacent
Cos A =h
a
Cos C =14
6
Cos C = 0.4286

6. Find angle x2.
8 cm
3 cm
x
a
o
Given adj and opp
need to use tan:
Tan A = adjacent
opposite
x = tan
-1
(2.6667)
x = 69.4
o
Tan A =a
o
Tan x =3
8
Tan x = 2.6667

7. Cos 30 x 7 = k
6.1 cm = k
7 cm
k
30
o
3. We have been given
the adj and hyp so we
use COSINE:
Cos A = hypotenuse
adjacent
Cos A =h
a
Cos 30 =7
k
Finding a side from a triangle

8. Tan 50 x 4 = r
4.8 cm = r
4 cm
r
50
o
4.
Tan A =a
o
Tan 50 =4
r
We have been given the
opp and adj so we use
TAN:
Tan A =

9. 45°-45°-90°Triangle
Theorem
•In a 45°-45°-90°
triangle, the
hypotenuse is √2
times as long as each
leg.x
x x√2
45°
45°
Hypotenuse = √2 * leg

10. 30°-60°-90°Triangle
Theorem
•In a 30°-60°-90°
triangle, the
hypotenuse is twice
as long as the shorter
leg, and the longer leg
is √3 times as long as
the shorter leg.
x√3
60°
30°
Hypotenuse = 2 ∙ shorter leg
Longer leg = √3 ∙ shorter leg2x
x

11. Ex. 1: Finding the hypotenuse in a 45°-
45°-90°Triangle•Find the value of x
•By the Triangle Sum
Theorem, the measure of
the third angle is 45°.
The triangle is a 45°-
45°-90°right triangle,
so the length x of the
hypotenuse is √2 times
the length of a leg.
3 3
x
45°

12. Ex. 1: Finding the hypotenuse in a 45°-
45°-90°Triangle
Hypotenuse = √2 ∙ leg
x = √2 ∙ 3
x = 3√2
3 3
x
45°
45°-45°-90°Triangle
Theorem
Substitute values
Simplify

13. Ex. 3: Finding side lengths in a 30°-
60°-90°Triangle•Find the values of s
and t.
•Because the triangle
is a 30°-60°-
90°triangle, the
longer leg is √3
times the length s
of the shorter leg.5
s
t
30°
60°

14. Ex. 3: Side lengths in a 30°-60°-90°
Triangle
Statement:
Longer leg = √3 ∙ shorter leg
5 = √3 ∙ s
Reasons:
30°-60°-90°Triangle
Theorem
5
√
3
√3
s√
3
=
5
√
3
s=
5
√
3
s=
√3
√
35√
33
s=
Substitute values
Divide each side by √3
Simplify
Multiply numerator and
denominator by √3
Simplify5
s
t
30°
60°

15. The length t of the hypotenuse is twice the length s of the shorter
leg.
Statement:
Hypotenuse = 2 ∙ shorter leg
Reasons:
30°-60°-90°Triangle
Theorem
t2
∙
5√
33
= Substitute values
Simplify5
s
t
30°
60°
t
10√
33
=