wynberg girls high-louise keegan-maths-grade11-trigonometry revision
DESCRIPTION
Revision of trigonometryTRANSCRIPT
Mathematics Revision
Gr 11 Trigonometry
Trigonometric Ratios
r
yA sin
r
xA cos
x
yA tan
y- axis
x- axis
P (x:y)
A
r
CAST Diagram
ALL
COS
SIN
TAN
Example 1
A
(-4; 3)
y
x
Work out sinA, cosA and tanA
Example 2
If and , evaluate:
1.
2.
03sin5 0tan
cossin
cos5tan4
Example 3
If where and
Determine, using a diagram, an expressionin terms of p for:
1.
2.
5cos
p 0p ]360;180[
tan
sin
cos
Steps to help you
• Make sure your ratio is in ‘simple form’,
i.e.
• Check in which quadrant the point P lies
• Draw a sketch
• Calculate unknown value (x, y, or r) using Pythagoras
• Substitute values into problem and solve
fractionsin
Special Angles
You must be able to calculate the values of ratios of the following angles without using a calculator:
0º, 30º, 45º, 60º, 90º, 180º, 270º, 360º
I remember them by using special triangles,
and my knowledge of graphs.
0º, 90º, 180º, 270º, 360º
30º, 45º, 60º
Special Triangles:
LEARN THESE!
Trig ratios (Using Triangles)
hyp
oppA sin
hyp
adjA cos
adj
oppA tanA
Hypotenuse
Adjacent
Opp
osite
Special Angles: Quick Test
Find the values of the following (no calculator)
)30sin45(cos30tan
)30(cos
60sin2
0cos270sin0sin
0cos90sin
45tan2
60cos30sin
2
2
12
234
3
34
2
2
2
1
Reduction Formulae 1:
Angles bigger than 360º:
sin(360º + A) = sinA
cos(360º + A) = cosA
tan(360º + A) = tanA
Reduction Formulae 2:
Negative Angles:
tan)tan(
cos)cos(
sin)sin(
x
yr
xr
y
Ө
Reduction Formulae 3:
(180º - Ө), (180º + Ө), (360º - Ө)
http://www.mindset.co.za/resources/0000013429/0000018906/0000033287/Reduction.htm
Reduction Formulae 3:
• (90º ± Ө)
Ө
(x,y)
sin)90cos(
cos)90sin(
r
yr
x
sin)90cos(
cos)90sin(
r
yr
x
sin and cos are co-functions
Reduction Formulae Summary:
Sin All
(180º - Ө) Ө (Ө ± 360º)
(90º + Ө) (90º - Ө)
Tan Cos
(180º + Ө) (360º - Ө)
- Ө
Examples:
2
3
225cos.104sin
)135sin(.14cos.300sin.480tan : thatProve
x)-x).sin(360-sin(180tan45
x)-cos(901 :Simplify
)tan().90sin().360cos(
)90sin().180tan().450sin( :Simplify
.sin300tan150
1330cos :Simplify
135tan150sin
)tan(-315-cos300 :Simplify
2
xxx
xxx
Trig Identities:
Quotient Identity:
Proof:
Squares Identity:
Proof:
tancos
sin
1cossin 22
y
x
r
Identities examples:
Prove the following identities:
1.
2.
3.
4.
5.
tansin
cos
cossin
sin21
sin
1
sin
cos
cos1
sincos
1costansin
1)tan1)(sin1(
tansin1
cos1
2
22
22
2
xx
x
x
xx
xxx
Calculator Work:NB: Always check that your calculator is on DEGREES.
1. sin 73° = 0.956304756…2. tan56° = 1,482560969…3. cos320° = 0,766044443…4. 2 sin20° = 2 x sin20° = 0,684040286…5. 2 sin30° + 3 cos10° = 3,954423259…6. tan (21° + 36°) = 1,539864964…7. tan21° + tan260° = 6,055145855…8. sin197°cos13° = -0,284878236… 9. sin2200° = 0,116977778…
Calculator Work: Equations
• sinθ = 0,823
θ = 55,386…
θ = 55,4°
• cosθ = 0,5
θ = 60°
• tanθ = 1,6003
θ = 57,9994…
θ = 58°
• 2 cos θ = 1,264
cos θ = 0,632
θ = 50,8°
Using your calculator to find the reference angle:
Equations
Consider the first example on the previous page sinθ = 0,823
Where can you find the solution off the graph?
Can you see that there are multiple solutions for θ
General vs. Specific Solution
• The general solution gives all the possible solutions for θ.
Remember:• sinθ and cosθ repeat every 360º (+ k.360º)• tanθ repeats every 180º (+ k.180º)
• The specific solution gives solutions for a restricted section.
• E.g. θ Є [-180; 180]
Equations examples
• sinθ = 0,823
ref L = 55,4°
• cosθ = 0,5
ref L = 60°
• tanθ = -1,6003
ref L = 58°
• -2 cos θ = 1,264
cos θ = -0,632
ref L = 50,8°
Find the general solutions for the following equations:
More complex equations eg.s:
Without using a calculator:1.
2.
More advanced:
1. and
2. with
2sin
12
1cos
x
]180;180[
32cos2 02tan
583,1)15cos(3 ]180;180[
Steps to help you
• Find the reference angle (Always +)
• Consider where the solution can be (Quad)
• Manipulate reference angle so that it satisfies given info
• Remember to add k.360º (sin & cos) or k.180º (tan).
• For specific solutions:
Choose answers that satisfy restriction.
sin, cos and area rules
• Sine Rule: (Given: 1 side and 2 angles or 2 sides and 1 angle)
• Cosine Rule: (Given 3 sides, or 2 sides and included angle)
• Area Rule: (Need 2 sides and included angle)
c
C
ba
A sinsinsin
B
Abccba cos2222 bc
acbA
2cos
222
CabArea sin2
1
Example 1: (sin rule)
Calculate c.
A
CB
76º
b = 5,5 cm
Example 2: (cos rule)
Calculate the magnitude of LEDF.
D
FE
9 cm
15 cm
8 cm
Example 3: cos rule
Calculate the length of PR.
P R
Q
120º
5 cm 3 cm
Example 4: area rule
Find the area of Δ XYZ:
45º
12 m
14 m
Extra examples:
cos and sine rule egs.pdf
gr 11 text book sincosarea.pdf
Trig Functions
Terms to know/understand:•Period - wavelength: time it
takes for the wave to complete a cycle
•Amplitude - displacement of the wave: i.e. the height the waves from its centre
•Asymptote - dotted line showing where the function may not touch
•Domain - x-values of function
•Range - y-values of function
Transformations on functions:
qpxy )sin(
)sin(kxay
Translation
Enlargement
Reflection
+: Moves up -: Moves down
+: Moves left -: Moves right
Stretches y-valuesi.e. changes amplitude
Stretches x-valuesi.e. changes period
k
360Period
a<0 Reflects over x-axis