Trigonometry : Trigonometry Right Angled Triangles
By Noel Platt
The sides of a Right Angled Triangle : The sides of a Right Angled Triangle X is the ANGLE.
The side OPPOSITE angle X is the OPPOSITE.
The LONGEST side is the HYPOTENUSE.
The side NEXT TO (ADJACENT to) angle X is the ADJACENT.
EXAMPLES OF OPPOSITE SIDES : EXAMPLES OF OPPOSITE SIDES
LABEL THE SIDES : LABEL THE SIDES QR
= OPPOSITE
PR
= HYPOTENUSE
QP
= ADJACENT
LABEL THE SIDES : LABEL THE SIDES ML
= OPPOSITE
MN
= HYPOTENUSE
NL
= ADJACENT
LABEL THE SIDES : LABEL THE SIDES EF
= ADJACENT
GE
= HYPOTENUSE
GF
= OPPOSITE
THE SINE RATIO : THE SINE RATIO SINE (X) = OPPOSITE/HYPOTENUSE
SIN(X) = 0/H
THE COSINE RATIO : THE COSINE RATIO COSINE (X) = ADJACENT /HYPOTENUSE
COS(X) = A/H
THE TANGENT RATIO : THE TANGENT RATIO TANGENT (X) = OPPOSITE/ADJACENT
TAN(X) = O/A
MNEMONIC : MNEMONIC Soh Cah Toa
Sin (x) = 0/H
Cos (x) = A/H
Tan (x) = O/A
X = THE ANGLE.
FORMULA TRIANGLES : FORMULA TRIANGLES The COVER UP RULE:
To find O (=opposite) in terms of the others COVER IT UP.
Read off O as Sin (x) times H (=HYPOTENUSE)
Etc.
FORMULA TRIANGLES : FORMULA TRIANGLES The COVER UP RULE:
To find A in terms of the others COVER IT UP.
A = Cos (x) times H.
Etc.
FORMULA TRIANGLES : FORMULA TRIANGLES The COVER UP RULE:
To find O (= opposite) in terms of the others, COVER IT UP.
O = Tan (x) times A
APPROPRIATE USE : APPROPRIATE USE If you are given:
A SIDE
and
AN ANGLE other than the right angle,
you can find
ANOTHER SIDE.
SA S
METHOD. : METHOD. To find a side,
given
another side and an angle:
1) Draw a labelled diagram
2) Write down SoH Cah Toa (the Mnemonic)
3) Use the Mnemonic and your GIVENS and GOAL to CHOOSE which ratio to use
4) Write down the formula triangle for the ratio you have chosen
5) Use your formula triangle to write down an equation to solve for the missing side.
6) Solve the equation.
EXAMPLE 1 : EXAMPLE 1 Find CD
Required to Find CD (RTF CD)
1) Label sides
CD = Opposite
CE = Adjacent
DE = Hypotenuse
EXAMPLE 1 : EXAMPLE 1 RTF CD:
2) Soh Cah Toa
3) Goal is to find CD = opposite (O)
Given =10cm = adjacent (A)
Choose
TOA which has O and A
EXAMPLE 1 : EXAMPLE 1 4) From formula triangle,
5) O = Tan (15) x A
6) O = 0.2679 x 10
O = 2.679 cm
EXAMPLE 2 : EXAMPLE 2 Find NM
1) RTF NM = Opposite
Given NL = 12 = HYPOTENUSE
& angle 32 degrees
LM = Adjacent whose length is NOT GIVEN
EXAMPLE 2 : EXAMPLE 2 2) Soh Cah Toa
3) Goal is to find NM = opposite (O)
Given =12cm = Hypotenuse (H)
Choose
SOH which has O and H
EXAMPLE 2 : EXAMPLE 2 4) From formula triangle,
5) O = Sin (32) x H
6) O = 0.5299 x 12
O = 6.359 cm
EXAMPLE 3 : EXAMPLE 3 Find RQ
1) RTF RQ = Adjacent
Given PQ = Hypotenuse = 34.3 cm
& angle = 50 degrees
PR = Opposite length is NOT GIVEN.
EXAMPLE 3 : EXAMPLE 3 2) Soh Cah Toa
3) Goal is to find RQ = Adjacent (A)
Given =34.3 cm = Hypotenuse (H)
Choose
CAH which has A and H
EXAMPLE 3 : EXAMPLE 3 4) From formula triangle
5) A = Cos (50) x H
6) A = 0.6428 x 34.3
A = 22.048 cm.
Finding side questions : Finding side questions 1) Find AC if x = 30 degrees, & AB = 15 cm
2) Find BC if x = 20 degrees & AC = 20 cm
3) Find BC if x = 12 degrees & AB = 18 cm
4) Find AB if x = 40 degrees & AC = 25 cm
Finding sides answers : Finding sides answers Solutions:
1) AC = 15 x Cos (30) = 12.99 cm.
2) BC = 20 x Tan (20) = 7.279 cm
3) BC = 18 x Sin (12) = 3.742 cm
4) AB = 25/Cos(40) = 32.635 cm
FINDING ANGLES. : FINDING ANGLES. To find angles from the lengths of two sides, you need to:
1) Divide the lengths of the two sides to obtain the appropriate Trigonometric ratio for the angle.
2) Find the angle by applying the INVERSE TRIGONOMETRIC FUNCTION to the RATIO.
FINDING ANGLES. : FINDING ANGLES. Example 1:
If BC = 7cm and AC = 12 cm Find angle x
BC = Opposite (0), AC = Adjacent (A)
Soh Cah Toa
FINDING ANGLES. : FINDING ANGLES. Choose Toa
Tan(x) = O/A = 7/12 = 0.5833
Angle x = Inverse Tan (0.5833) =30.256 degrees
FINDING ANGLES. : FINDING ANGLES. Example 2:
If BC = 3 cm and AB = 5 cm Find angle x
BC = Opposite (0), AB = Hypotenuse (H)
Soh Cah Toa
FINDING ANGLES. : FINDING ANGLES. Choose Soh
Sin(x) = O/H = 3/5 = 0.6
Angle x = Inverse Sin (0.6) =36.87 degrees
FINDING ANGLES. : FINDING ANGLES. Example 3:
If AB = 16 cm and AC = 5 cm Find angle x
AB = Hypotenuse (H), AC = Adjacent (A)
Soh Cah Toa
FINDING ANGLES. : FINDING ANGLES. Choose Cah
Cos(x) = A/H = 5/16 = 0.3125
Angle x = Inverse Cos (0.3125) =71.79 degrees
FINDING ANGLES QUESTIONS. : FINDING ANGLES QUESTIONS. 1) Find angle x if FG = 16ft and EF = 40 ft
2) Find angle x if EG = 19.5 cm and EF = 14 cm.
3) Find angle x if EG = 5 cm and FG = 4 cm
4) Find angle x if EF = 30 m and FG = 15 m
FINDING ANGLES ANSWERS. : FINDING ANGLES ANSWERS. SOLUTIONS:
1) Angle x = Inverse Tan (16/40) =21.8 degrees
2) Angle x = Inverse Cos (14/19.5) = 44.115 degrees
3) Angle x = Inverse Sin (4/5) =53.13 degrees
4) Angle x = Inverse Tan (15/30) = 26.565 degrees.