trig. graphs and equations -...

Trig. graphs and equations December 11, 2014

Today we will be learning about Trig. Graphs

y = Sinx0

Trigonometric Graphs

y = Cosx0


y = Tanx0



The amplitude of a graph = (Distance between max. and min.) ÷ 2

The period of a graph is the length of the graph before it repeats itself.

Period = 3600 ÷ b when y = asinbx0 and y = acosbx0 Period = 1800 ÷ b when y = atanbx0

36001800

a

Period

Trigonometric Graphs Reminder: Graph transformations

Advice: - Draw a new graph for each bit added to the function- Labels are expected.- If unsure about where it cuts x and y axis, find roots and y - intercept.


Trigonometric Graphs Examples: 1. Sketch and annotate the graph of the function

y = 2sin3x0 + 1

Trigonometric Graphs Examples: 2. Write a trigonometric function represented by the graph belowthat is in the form y = acos(x - b)0 + k (Not to scale)

3

-5

600 4200

Trigonometric Graphs Examples: 3. Sketch and annotate the graph of the function

y = 2sin(x0 + 30)+ 1

Daily Practice 20.11.14

Q1. State the turning point and y - intercept of the function y = (x + 3)2 - 2

Q2. State the equation of the line that passes through

(2, 3) and is parallel to the x - axis

Q3.On a suitable set of real numbers, functions f and g are defined f(x) = 2x and g(x) = sinx + cosx. Find in its simplest form an expression for g(f(x))

Today we will be learning about radians.

Homework Online due Tuesday 25.11.14

Radians

A radian is the angle at the centre of a sector of a circle where the length of the arc is equal to the radius.

r

r

r1 radian

Arc Length = x0 x πD360


Radians

To convert angles to radians:Angle x π

To convert radians to angles: just substitute 1800 in for π or multiply it by 1800

1800

π

Radians

Examples:1. Convert 2400 to radians

2. Convert 6π to degrees5

Daily Practice 24.11.14Q1.

Write down the centre and radius

Q2.

Sketch y = g'(x)

and

y = g'(x - 3)

Today we will be learning about exact values.

Homework Due Tomorrow!

Exact ValuesThese are the values for sin, cos and tan of 300, 450, 600 & 900 written in accurate form and should be known without the use of a calculator.

You can find them by using an equilateral triangle and a square

Exact Values

x 0 π

cosx

sinx

tanx


Exact Values

22

Exact Values

1

11

1

Exact Values

x 0 π

cosx 1 1 0

sinx 0 0 1

tanx 0 0 undefined 1


Q1. Given the function y = (2 - x)(x + 3), find the gradient and equation of the tangent to the function at the point (-1, 6)

Q2. Triangle ABC has vertices A(-3, -3), B(-1, 1) and C(7, -3). Show that the triangle ABC is right-angled.

Q3. The diagram shows part of the graph y = sinx0 and the graph of a related function. State the equation of the related function

y = sinx0

x

y1

0

-1

Today we are going to revise over the CAST diagram and solve trig. equations.

Homework Due!

Using the CAST diagram

The CAST diagram is a quick way of showing us the symmetry on trig. graphs.


1800 - x0 1800 + x0

3600 - x0 x0

Cosx0

00 , 3600

900

1800

2700

π2

3π2

(2π)(π)

1800 - x0

1800 + x0

3600 - x0

x0

Tanx0

00 , 3600

900

1800

2700

π2

3π2

(2π)(π)

1800 - x0

1800 + x0 3600 - x0

x0

Sinx0

00 , 3600

900

1800

2700

π2

3π2

(2π)(π)

x0 (Acute angle or reference angle)

00 , 3600

900

1800

2700

π2

3π2

(2π)(π)

Cos Positive

All PositiveSin Positive

Tan Positive

Using the CAST diagram

First identify which quadrant the angle lies in.

In order to find the exact value , rewrite it in terms of its associated acute angle first.

Positive angles are measured anticlockwise from 0 and negative angles are measured clockwise from 0. E.g. Cos(-1200)= cos2400

Exact Values

Examples:

1. Find the exact value of sin2250 x 0 π

cosx 1 1 0

sinx 0 0 1



Exact Values

Examples:

2. Find the exact value of tan(-300) x 0 π

cosx 1 1 0

sinx 0 0 1



Q1. For what value of k does the equation 2x2 + 4x + k = 0 has real roots?

Q2. Find the equation of the tangent to y = x3 - 2x2 + 4 at the point (2, 4)

Q3. The diagram shows the graph of f(x) = 2cosx0, sketch the graph of y= f(2x)

Exact Values

Examples:

3. Find the exact value of sin x 0 π

cosx 1 1 0

sinx 0 0 1


4. Find in its simplest form the exact value of 2 sin 210˚cos 330˚

Exact Values

Examples:

x 0 π

cosx 1 1 0

sinx 0 0 1


Today we will be learning to solve trig. equations.

Solving Trigonometric Equations

Trig. Equations can be solved using the graph (when y = ?, find the value of x)

For example: Solve the equation cosx0 = 0.5


Solving Trig. Equations Algebraically

Note: If the question has no degree symbol above x, then the answer is expected in radians.

Solve the equations like you would solve regular equations. Some may even involve factorising.

Solving Trig. Equations AlgebraicallyExamples:1. Solve 3sinx0 + 2 = 0 where (00 ≤ x ≤ 3600)

Solving Trig. Equations AlgebraicallyExamples:2. Solve 2sin2x0 - 1 = 0 where (00 ≤ x ≤ 3600) (From HSN notes)

Solving Trig. Equations AlgebraicallyExamples:3. Solve 4cos2x = 3 where (0 ≤ x ≤ 2π) (From HSN notes)

Solving Trig. Equations AlgebraicallyExamples:4. 3sin2x0 - 4sinx0 + 1 = 0

(From Heinemann book)


Q1. Show that the line with equation y = -3x + 6 is a tangent to the circle with equation x2 + y2 + 10x - 2y - 14 = 0

Q2. A sequence is defined by the recurrence relation un+1 = 0.3un + 5.

Explain why this sequence has a limit as n-> ∞ and find the exact value of this limit.

Q3. Write the the function y = 9x2 + 6x + 3 in completed square form and state the stationary point


Show that the line with equation y = -3x + 6 is a tangent to the circle with equation x2 + y2 + 10x - 2y - 14 = 0

Today we will be finishing off our practise of trig. equations and starting vectors.

trig. graphs and equations -...

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