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### Trigonometry

Right angled triangles

Opposite and Adjacent are relative to the angle

- The 4cm side is opposite to A
- The 6cm side is adjacent to A
- The 6cm side is opposite to B
- The 4cm side is adjacent to B

SOH CAH TOASine, Cosine & Tangent of an angle

SOH CAH TOA

- SOH
- Some Old Houses
- (Sine of angle =Opposite side / Hypotenuse)

- Some Old Houses
- CAH
- Creak And Howl
- (Cosine of angle = Adjacent side / hypotenuse)

- Creak And Howl
- TOA
- Through Out Autumn
- (Tangent of angle = Opposite side / adjacent side)

- Through Out Autumn

TOA SOH CAH

- TOA
- Tom’s Old Auntie
- Tangent of angle = Opposite side / adjacent side

- Tom’s Old Auntie
- SOH
- Sat On Him
- Sine of angle =Opposite side / Hypotenuse

- Sat On Him
- CAH
- Cursing At Him
- Cosine of angle = Adjacent side / hypotenuse

- Cursing At Him

SOH CAH TOA

- SOH

S

O

H

Opposite side = sin (x) times hypotenuse

Hypotenuse = opposite side divided by sin (x)

Sin (x) = opposite side divided by hypotenuse

SOH CAH TOA

C

A

- CAH

H

Adjacent side = cos (x) times hypotenuse

Hypotenuse = adjacent side divided by cos (x)

Cos (x) = adjacent side divided by hypotenuse

SOH CAH TOA

A

- TOA

T

O

Opposite side = tan (x) times adjacent side

Adjacent side = opposite side divided by tan (x)

tan (x) = opposite side divided by adjacent side

Problems for trig

A stage, is to be built for a concert, it has to be 2m high so the audience can see the show.

The equipment needs to be pushed up onto the stage. Health and safety rules say that a ramp must have a slope of no more than 15 degrees.

The crew need to work out how far away from the stage to start building the ramp.

m

A stage, is to be built for a concert, has to be 2m high so the audience can see the show.

The equipment needs to be pushed up onto the stage. Health and safety rules say that a ramp must have a slope of no more than 15 degrees.

The crew need to work out how far away from the stage to start building the ramp.

m

Put answer in context: the audience can see the show.

The ramp must start from at least 7.5m away from the stage

From take-off, an aeroplane climbs at an angle of 20 the audience can see the show. o. When the aeroplane has flown 10km, what height has it reached?

km

Distance from ground.

Looking for opposite (distance from ground) got hypotenuse. the audience can see the show. Must be Sine formula triangle

km

Distance from ground.

Distance from ground = sin (20) x 10

= 3.42 km from the ground

A plane flies 300km on a bearing of 132 the audience can see the show. 0 from an airport. How far south and east is it from the airport. Give answer correct to 3 s.f.

North is always straight up your page for these questions..

Bearings are always measured from north around in a clockwise direction.

Draw the problem and work out the angle at A the audience can see the show. Then choose which side you want to find first then choose the formula triangle to suit.

Distance South first the audience can see the show. looking for adjacent side therefore use cosine formula triangle

Distance East the audience can see the show. looking for opposite side therefore use the sine formula triangle

Ratio just means the number you get when you divide one number by another

Similar shapes have the same angles – so they have the same angle ratios.

Sin(30) is always the same number no matter what size the opp or hyp

A

Cos(30) is always the same number no matter what size the adj or hyp

A1

A2

Tan(30) is always the same number no matter what size the opp or adj

B

C

C1

C2

Sine ratio number by anotherSin(x)=opp / hyp

4.3 / 8.42 = ?

To find the angle (x) when you know sin (x) use the calculator inverse sine function (sin-1)

- Sin (x) = 0.86
x = Sin-1 (0.86) = 59 0

- Sin (x) = 0.35
x = Sin-1 (0.35) = 21 0

Sin (x) = 0.45 , what is x ?

Sin (x) = 0.91, what is x?

27 0

66 0

To find the angle (x) when you know cos (x) use the calculator inverse cosine function (cos-1)

- Cos (x) = 0.86
x = Cos-1 (0.86) = 31 0

- Cos (x) = 0.35
x = Cos-1 (0.35) = 70 0

Cos (x) = 0.45 , what is x ?

Cos (x) = 0.91, what is x?

63 0

24 0

To find the angle (x) when you know tan (x) use the calculator inverse tangent function (tan-1)

- tan (x) = 0.86
x tan(0.86) = 41 0

- tan (x) = 0.35
x = tan-1 (0.35) = 19 0

tan (x) = 0.45 , what is x ?

tan (x) = 0.91, what is x?

24 0

42 0

A tourist lift to the top of a cliff travels 23m from ground to the top of the cliff. The height from ground to the top of the cliff is 20m what is the angle of elevation?

Choose a formula triangle

Sketch the problem

We have opposite and hypotenuse must be sine formula triangle

Looking for the angle BAC. to the top of the cliff. The height from ground to the top of the cliff is 20m what is the angle of elevation?Work out the sine of the angle BAC then use inverse sine to get angle.

Health and Safety stipulates that a ladder held up at the side of a wall must have an angle of elevation between 700 and 800 to be considered safe. The height to be reached is 2.4m but the only ladder available is 4.9m will it be classed as safe?

Not safe

Find angles A and B? side of a wall must have an angle of elevation between 70We know all sides so ANY formula triangle will be ok to use.

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