 # Trigonometry.

## Presentation on theme: "Trigonometry."— Presentation transcript:

Trigonometry

Trigonometry is a method of finding out an unknown angle or side in a right angled triangle
Both the triangles below are similar because: The angles are the same but the sides are different

Trigonometry is a method of finding out an unknown angle or side in a right angled triangle
Both the triangles below are similar because: The angles are the same but the sides are different

Trigonometry is a method of finding out an unknown angle or side in a right angled triangle
Both the triangles below are similar because: The angles are the same but the sides are different

If we measure the height and the base:
Small triangle Large triangle For both triangles 5 cm 8 cm 10 cm 16 cm

This angle is in fact 320 So as long as the value of then this angle will always be 320

This is the idea behind trigonometry
If we know 2 sides then we can find the angles in the triangle How do we know the angle is 320 ? We can use our calculator which has been programmed to work out the angle.

We don`t have to know the height and the base it can be any 2 sides
Depending on which 2 sides are known then we use a different button on the calculator Names are given to the 3 sides which all refer to the angle we are trying to find

Opposite, Adjacent and Hypotenuse
The names are: Opposite, Adjacent and Hypotenuse Opposite means on the other side from the angle we need. Adjacent means next to the angle we need. Hypotenuse means the side opposite the right angle Hypotenuse Opposite X Adjacent

Identify the names of the sides of these right angled-triangles given angle k
opposite b opposite a b hypotenuse c c adjacent a hypotenuse k opposite a k adjacent c adjacent b c opposite k hypotenuse hypotenuse b k a adjacent

Opposite, Adjacent and Hypotenuse
In each case label all the sides of the triangles as Opposite (O), Adjacent (A) and Hypotenuse (H) with relation to the angle marked as “X”. X X X x X X X X X X X

Using the Opposite (O), Adjacent (A) and
Hypotenuse (H) to work out the missing angle The calculator has 3 buttons which are used to find the missing angle: Sin – short for Sine Cos – short for Cosine Tan – short for Tangent

Deciding which button to use depends on which sides are given
SOH CAH TOA Memory Aid Some Old Horses Sin Opposite Hypotenuse Can Always Hear Cos Adjacent Hypotenuse Their Owners Approaching Tan Opposite Adjacent Or invent one of your own

SOH CAH TOA S O H C A H T O A Divide it up into three groups
Place each group of three in a triangle starting in the bottom left of each triangle S O H C A H T O A

Trigonometric Ratios S O H SOH C A H CAH T O A TOA

SOH CAH TOA Cos (x) = x Example 1 What have we got and need to find?
We need an angle – x. We have the Hypotenuse and Adjacent side. 10 cm 25 cm x Looking at the phrase, we can use C A H Hypotenuse Cos (x) = Adjacent Hypotenuse Adjacent

Cos (x) = Adjacent Hypotenuse 10 cm 25cm x Replace A and H by 10 and 25 Hypotenuse Cos (x) = = 0.4 We now need to convert this to an angle in degrees using the Cos-1 button!!! Adjacent x = Cos –1(0.4) = 66.42o We always find the angle using either the Cos–1, Sin–1 or Tan–1 buttons.

SOH CAH TOA Tan (x) = x Example 2 What have we got and need to find?
We need an angle – x. We have the Opposite and Adjacent side. Looking at the phrase, we can use TOA 15 cm Tan (x) = Opposite Adjacent Opposite x 20 cm Adjacent

We always find the angle using either the Cos–1, Sin–1 or Tan–1 buttons.
Tan (x) = Opposite Adjacent Replace O and A by 15 and 20 Tan (x) = = 0.75 We now need to convert this to an angle in degrees using the Tan-1 button!!! 15 cm Opposite x x = Tan –1(0.75) = 36.67o 20 cm Adjacent

SOH CAH TOA Sin (x) = x Example 3 What have we got and need to find?
We need an angle – x. We have the Hypotenuse and Opposite side. Looking at the phrase, we can use S O H Hypotenuse 12 cm 8 cm Sin (x) = Opposite Hypotenuse Opposite x

Sin (x) = Opposite Hypotenuse We always find the angle using either the Cos–1, Sin–1 or Tan–1 buttons. Replace O and H by 8 and 12 Sin (x) = = 0.666 Hypotenuse We now need to convert this to an angle in degrees using the Sin-1 button!!! 12 cm 8 cm Opposite x x = Sin –1(0.666) = 41.81o

Using Trigonometry to Find a Missing Side

Trigonometric Ratios S O H SOH C A H CAH T O A TOA

Trigonometric Ratios S O H
The triangle can also be used to find either the opposite side or the hypotenuse

Trigonometric Ratios C A H
The triangle can also be used to find either the adjacent side or the hypotenuse

Trigonometric Ratios O A T
TOA The triangle can also be used to find either the adjacent side or the opposite

H S O H Example 1 SOH CAH TOA What have we got and need to find?
We need the Hypotenuse H We have an angle and the Opposite O Hypotenuse Looking at the phrase we can use S O H S O H Opposite Hypotenuse = Opposite Sin (angle)

H Opposite Hypotenuse = Sin (angle) Replace O by 3 and (angle) by 60o
Sin (60o) 60o Use the Sin button on your calculator to find this value H H = Hypotenuse H = ….. H = 3.46 m to 2 d.p. 3 m Opposite

A T O A Example 2 SOH CAH TOA What have we got and need to find?
We need the Adjacent A 40o We have an angle and the Opposite O A Adjacent Looking at the phrase we can use T O A T O A 3 m Opposite

A Replace O by 3 and (angle) by 40o 3 40o Tan (40o) H = Adjacent H =
Use the Tan button on your calculator to find this value A H = Adjacent H = 3.575….. 3 m H = 3.58 m to 2 d.p. Opposite

8 C A H Example 3 SOH CAH TOA What have we got and need to find?
We need the Adjacent A We have an angle and the Hypotenuse H 8 Hypotenuse Looking at the phrase we can use C A H 70o A C A H Adjacent

8 Replace H by 8 and (angle) by 70o A = cos70 x 8 H = 0.342 x 8
Use the Cos button on your calculator to find this value 8 H = x 8 Hypotenuse H = 2.736….. 70o A H = 2.74 m to 2 d.p. Adjacent