working out an unknown side or angle in a right angled triangle. miss hudson’s maths

Working out an unknown side or angle in a right angled triangle.

TRIGONOMETRY

Miss Hudson’s Maths

SOH CAH TOA

The sides of a right angled triangle are named according to their position relative

to the angle

Hypotenuse (Hyp)

Opposite (Opp)

Adjacent (Adj)


PR

AC

TIC

E

LA

BELIN

G A

TR

IAN

GLE!


The sides are paired up to form the three ratios called sin, cos and tan (sine, cosine, tangent).

adj

opptan

hyp

adjcos

hyp

oppsin

SOHCAHTOA

is often used to help remember

these ratios

(Hyp) (Opp)

(Adj)

SOHCAHTOAis used to help

remember these ratios

soh

ah

c toa

O

H

A


CALCULATING AN UNKNOWN SIDE

METHOD:

1. Label sides with information on them as ‘O’, ‘A’ or ‘H’

2. Use SOHCAHTOA to help choose which ratio to use: sin, tan or cos.

3. Write down the ratio

4. Put in the values from the question

5. Solve for the unknown side.

OH

eg 1: Find the length of the side labelled y

y10 cm

65°

sin 65°= 10

y

y = 9.06 cm (2dp)

y = 10 x sin 65

10 x x 10

OH

sin 65°= HO


eg 2: Find the length of the side k

78°

k

3.2m

O

A

tan 78°=2.3

k x 3.2 3.2 x

k = 3.2 x tan 78

k = 15.05 m (2dp)

Make sure your

calculator says ‘D’ and not ‘R’ or ‘G’

SOHCAHTOA

eg 3: Find the length of the side labelled y

A

H

cos 49° =y

5.6cm

49° 6.5

y x 5.65.6 x

y = 5.6 x cos 49y = 3.67 cm (2dp)

OAAH

tan 78°=A

O

cos 49° = H

A


eg 4: Find the length of the side labelled p

It’s not THAT

hard Bart!

42°

p7.2 m sin 42°

=O

H p

2.7

Since the unknown is on the denominator, interchange as shown

p = 42sin

2.7

p = 10.76 m (2 dp)

sin 42° =

H

O


CALCULATING AN UNKNOWN ANGLEMETHOD:

1. Label sides with information on them as ‘O’, ‘A’ or ‘H’

2. Use SOHCAHTOA to help choose which ratio to use: sin, tan or cos.

3. Write down the ratio

4. Put in the values from the question

5. Use SHIFT then the ratio to find the angle

7 cm

4 cm

eg 1: Find the size of angle

A

H

AH

cos =7

4

= cos -1 ( )7

4

SHIFT COS 4 7ab/c =

= 55.2° (1 dp)

To work out the angle use cos inverse, ie cos-1

cos =H

A


Angle of Elevation:

This is the angle measured from the horizontal UP to somethingeg:

angle of elevation

Angle of Depression:

This is the angle measured from the horizontal DOWN to something

eg: angle of depression


VECTORSA vector is a quantity which has length and direction.

A vector is described as where x is the distance across to

the right (+) or left (-) and y is the distance up (+) or down (-)

xy

A vector is often labelled with a small letter, either in bold type or in ordinary type with a marking under it, ie a or a

eg: a =

41

eg: b =

-2-3

a

b

eg: c =

3-5

c

Adding VectorsTo add vectors geometrically, we draw the 1st vector and then draw the 2nd vector starting at the end of the 1st one. 4

13-5

a b

a + b

eg: If a = and b = then a + b is the

shortest route from the very start to the end and is called the resultant.

a + b =

7-4

NB: + =41

3-5

7-4

+

+

Multiplying VectorsIf a =

12

then 3a = 3 12

= 3x1 3x2

= 36

a 3a

NB: 3a is 3 times as long as a and is parallel to it.

Magnitude & Direction

The direction of a vector is its angle. This is calculated using SOHCAHTOA.

The magnitude of a vector is its length. It is written as, for example, |a| This is calculated using Pythagoras’ Theorem.

eg: If b =

34

b3

4

|b|2 = 32 + 42

|b|2 = 25|b| = √25|b| = 5

Let θ be the angle the vector makes with the horizontal.

θ

tan θ = 3

4

θ = tan-1 ( )3

4

θ = 53.1° (1 dp)

direction is 53.1° from the horizontal length is 5 units

working out an unknown side or angle in a right angled triangle. miss hudson’s maths

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