Download Finding an angle from a triangle

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 ratios of two sides
of the triangle.
hyp
opp
θ
The sides of the right triangle are:
 the side opposite the acute angle ,
 the side adjacent to the acute angle ,
 and the hypotenuse of the right triangle.
adj
Right Triangle Trigonometry
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 90o.
A
A
Trigonometric Ratios
hyp
opp
θ
adj
The trigonometric functions are:
sine, cosine, tangent, cotangent, secant, and cosecant.
sin  = opp
cos  = adj
tan  = opp
hyp
hyp
adj
csc  = hyp
opp
sec  = hyp
adj
cot  = adj
opp
S OH C AH T OA
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.
1.
Find angle C
14 cm
C
6 cm
a) Identify/label the names of
the sides.
b) Choose the ratio that
contains BOTH of the
letters.
1.
We have been given the
adjacent and hypotenuse so
we use COSINE:
h
14 cm
adjacent
Cos A = hypotenuse
C
6 cm
a
a
h
Cos C = 6
14
Cos A =
Cos C = 0.4286
C = cos-1 (0.4286)
C = 64.6o
2. Find angle x
Given adj and opp
need to use tan:
x
3 cma
Tan A =
o 8 cm
Tan A =
o
a
Tan x =
8
3
Tan x = 2.6667
x = tan-1 (2.6667)
x = 69.4o
opposite
adjacent
Finding a side from a triangle
3.
7 cm
k
30o
We have been given
the adj and hyp so we
use COSINE:
adjacent
Cos A =
hypotenuse
Cos A = a
h
Cos 30 = k
7
Cos 30 x 7 = k
6.1 cm = k
4.
We have been given the opp
and adj so we use TAN:
50o
4 cm
Tan A =
r
Tan A =
Tan 50 =
Tan 50 x 4 = r
4.8 cm = r
o
a
r
4
45°-45°-90° Triangle Theorem
• In a 45°-45°-90°
triangle, the
hypotenuse is √2
times as long as each
leg.
45°
x√2
x
45°
x
Hypotenuse = √2 * leg
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.
60°
2x
x
30°
x√3
Hypotenuse = 2 ∙ shorter leg
Longer leg = √3 ∙ shorter leg
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
45°
x
Ex. 1: Finding the hypotenuse in a 45°-45°90° Triangle
3
3
45°
Hypotenuse = √2 ∙ leg
x
45°-45°-90° Triangle
Theorem
x = √2 ∙ 3
Substitute values
x = 3√2
Simplify
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.
60°
t
s
30°
5
Ex. 3: Side lengths in a 30°-60°-90° Triangle
60°
t
s
30°
5
Statement:
Reasons:
Longer leg = √3 ∙ shorter leg
5 = √3 ∙ s
5 = √3
s√
√
35
3
= s
√
√3 35 = s
√ √
3 35√
= s
3
30°-60°-90° Triangle Theorem
Substitute values
Divide each side by √3
Simplify
Multiply numerator and
denominator by √3
Simplify
The length t of the hypotenuse is twice the length s of the shorter leg.
60°
t
s
30°
5
Statement:
Reasons:
Hypotenuse = 2 ∙ shorter leg
30°-60°-90° Triangle Theorem
t = 2 5√
∙ 33
Substitute values
t = 10√
33
Simplify