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

A C

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.

RATIOSm A SIN A COS A TAN A0 0 1 020 .342 .9397 .36430 .5 .866 .577445 .7071 .7071 160 .866 .5 1.732180 .9848 .1736 5.671390 1 0

SIN 30 = COS 30 TAN 30 = = SIN 60 COS TAN 60SIN 45 COS 45 TAN 45

SINE

SINE A =

SINE A = C

B

A

ca

b

COSINE

COS A =

COS A =

B

A

C

ac

b

TANGENT

TAN A =

TAN A =

B

A

C

ca

b

CALCULATOR

CALCULATOR:

The button (key) SIN on the calculatorenables you to calculate the value of SIN A if you know the measurement ofANGLE 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 Aie. () = 30

A

B

C

ANGLES / SIDESFINDING MISSING SIDES USING TRIGONOMETRIC RATIOSIN 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 cm⁰4x A

B C

4 cm30⁰

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 = 53.1⁰

54 )

54(

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 ¯ = 41.1 ¹ ⁰43 )

43(

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 ¹ ⁰23

)23(

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 LAWThe 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

B CH a

bch

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 cm36.3

117.3

26.4

6 cm

12 cm

8 cm

A = = 21.3

P =

A = A = = 21.3

A =

A =

A =

a

bcA

B C