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TRIGONOMETRYBASIC TRIANGLE STUDY:
RATIOS:
- SINE
- COSINE
- TANGENT
- ANGLES / SIDES
SINE LAW:
AREA OF A TRIANGLE:
- GENERAL
- TRIGONOMETRY
- HERO’S
BASIC TRIANGLE STUDY
Complimentary angles: 2 angles = 90
Supplementary angles: 2 angles = 180
Adjacent angles on the same line = 180
Opposite angles on the same line = each other
The sum of the interior angles of a triangle = 180
Right triangles have one angle = 90
Pythagorean Theorem = a² + b² = c²
RATIOS
“SOH” – “CAH” – “TOA”
TRIGONOMETRIC RATIOS IN A RIGHT TRIANGLE
SINE A = =
COSINE A = =
TANGENT A = =
c
b
a
AC
B
The adjacent side is the side next to the reference angle.The opposite side is the side directly across from the reference angle. Remember, it is important to understand that the names of the opposite side and adjacent sides change when you move from one reference angle to the other.
RATIOS
m A SIN A COS A TAN A
0 0 1 0
20 .342 .9397 .364
30 .5 .866 .5774
45 .7071 .7071 1
60 .866 .5 1.7321
80 .9848 .1736 5.6713
90 1 0
SIN 30 = COS 30 TAN 30 = =
SIN 60 COS TAN 60
SIN 45 COS 45 TAN 45
SINE
SINE A =
SINE A = C
B
A
ca
b
COSINE
COS A =
COS A =
B
A
C
a
c
b
TANGENT
TAN A =
TAN A =
B
A
C
ca
b
CALCULATOR
CALCULATOR:
The button (key) SIN on the calculator
enables you to calculate the value of
SIN A if you know the measurement of
ANGLE A.
ie. SIN 30 = 0.5
The button (key) on the calculator enables you to calculate the measure of the ANGLE A if you know SIN A
ie. () = 30
A
B
C
ANGLES / SIDESFINDING MISSING SIDES USING TRIGONOMETRIC RATIOS
IN A RIGHT TRIANGLE,
Finding the measure of x of side BC opposite to the known ANGLE A, knowing the measure of the hypotenuse, requires the use of SIN A.
SIN 50 = or x = 5 · SIN 50 = 3.83
Finding the measure of y of side AC adjacent to the known ANGLE A, knowing the measure of the hypotenuse, requires the use of COS A.
COS 50 = or y = 5 COS 50 = 3.21
5 cm
x
y
A
BC
50⁰
ANGLES / SIDES
FINDING MISSING SIDES USING TRIGONOMETRIC RATIOS IN A RIGHT TRIANGLE (CONTINUED),
Finding the measure of x of side BC opposite to the known angle A, knowing also the measure of the adjacent side to angle A, requires the use of TAN A
TAN 30 = ⁰ ⇒ x = 4 · TAN 30 = 2.31 cm⁰4
x A
BC
4 cm
30⁰
x
ANGLES / SIDES
FINDING MISSING ANGLES USING TRIGONOMETRIC RATIOS IN A RIGHT TRIANGLE,
Finding the acute angle A when its opposite side and the hypotenuse are known values require the use of SIN A.
SIN A = ⇒ m ∠A = SIN¯ ¹ = 53.1⁰5
4 )5
4(
A
B
C
4 5
ANGLES / SIDES
FINDING MISSING ANGLES USING TRIGONOMETRIC RATIOS IN A RIGHT TRIANGLE,
Finding the acute angle A when its adjacent side and the hypotenuse are know values require the use of COS A
COS A = ⇒ m ∠ A = COS ¯ = 41.4 ¹ ⁰4
3)4
3(
A
B
C
4
3
ANGLES / SIDES
FINDING MISSING ANGLES USING TRIGONOMETRIC RATIOS IN A RIGHT TRIANGLE,
Finding the acute angle A when its opposite side and adjacent side are known values requires the use of TAN A
TAN A = ⇒ m ∠ A = TAN ¯ = 56.3 ¹ ⁰2
3)2
3(
A
B
C
3
2
SINE LAW
The sides in a triangle are directly proportional to the SINE of the opposite angles to these sides.
The SINE LAW can be used to find the measure of a missing side or angle.
CASE 1: Finding a side when we know two angles and a side
We calculate the measure of x of AC
A
B Ca
bc
15 cm
x
A
B C50
60
SINE LAW
The SINE LAW can be used to find the measure of a missing side or angle.
CASE 2: Finding the angle when we know the two sides and the opposite angle to one of these sides
We calculate the measure of angle B
10 cm
13 cm
A
B C
50
x
AREA OF A TRIANGLE
GENERAL FORM
AREA =
or
AREA = BASE
HEIGHT L
W
AREA OF A TRIANGLE
TRIGONOMETRIC FORMULAAREA =
AREA =
AREA =
A
BC
H a
bc
h
AREA OF A TRIANGLE
HERO’S FORMULA When you are given the measures for all three sides a, b,
c of a triangle, Hero’s Formula enables you to calculate the area of a triangle.
AREA =
P = half the perimeter of the triangle
a
bc
A
B C
AREA OF A TRIANGLEGENERAL TRIGONOMETRIC
HERO’S
3.55 cm
12 cm
6 cm
8 cm
12 cm
36.3
117.3
26.4
6 cm
12 cm
8 cm
A = = 21.3
P =
A = A = = 21.3
A =
A =
A =
a
bc
A
B C
