S56 (5.3) Trig. Functions.notebook
Daily Practice
Q1.
December 17, 2015
7.12.15
Write down the
centre and radius
Q2.
Today we will be learning about radians.
Sketch y = g'(x)
Homework due tomorrow.
and
y = g'(x - 3)
Radians
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
Arc Length = x0 x πD
360
To get an answer in radians on your calculator, just use RAD mode.
To convert degrees to radians or radians to degrees, always remember
1800 = π radians
1 radian
r
Daily Practice
Radians
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)
Examples:
1. Convert 2400 to radians
2. Convert 6π to degrees
5
Ex. 4C
Q1-4
3. Evaluate cos 4π
3
8.12.15
Q2. Triangle ABC has vertices A(-3, -3), B(-1, 1) and C(7, -3). Show that
the triangle ABC is right-angled.
S56 (5.3) Trig. Functions.notebook
December 17, 2015
Exact Values
These are the values for sin, cos and tan of 300, 450 & 600 written in
accurate form and should be known without the use of a calculator.
Today we will be learning about exact values.
You can find them by using an equilateral triangle and a square
Homework Due!
Exact Values
Exact Values
x
0
π
cosx
sinx
2
2
tanx
1
1
Exact Values
Exact Values
1
1
1
x
0
π
cosx
1
­1
0
sinx
0
0
1
tanx
0
0
undefined
1
1
S56 (5.3) Trig. Functions.notebook
Daily Practice
December 17, 2015
9.12.15
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
Today we are going to learn about related
the point (2, 4)
angles.
Homework Online due 15.12.15
0
Q3. The diagram shows the graph of f(x) = 2cosx , sketch
the graph of y= f(2x)
2
3600
­2
Exact values: Related Angles
Cosx0
Due to symmetry in the graphs of sin, cos and tan, you can see that there
are a number of angles that have the same y value but might be negative.
x0
These are known as related angles.
3600 - x0
1800 - x0 1800 + x0
Tanx0
Sinx0
x0
x0
1800 - x0
1800 + x0
1800 - x0
3600 - x0
1800 + x0 3600 - x0
S56 (5.3) Trig. Functions.notebook
Exact values: Related Angles
We will note the related angles of 300, 450 and 600 because we
know their exact values.
December 17, 2015
Finding exact values of related angles
In order to find the exact value of a related angle:
Rewrite it in terms of its associated acute angle first.
x0
A negative angle is the same as (3600 - Angle).
E.g. Cos(-1200)= cos2400
1800 - x0
180 + x
0
0
3600 - x0
Adding 3600 wave(s) on will also give the same value.
e.g. sin300 and sin 7500
Exact Values
Exact Values
Examples:
Examples:
2. Find the exact value of tan(-300)
1. Find the exact value of sin2250
x
0
π
cosx
1
­1
sinx
0
0
1
0
undefine
d
tanx
Daily Practice
0
0
10.12.15
Q1. Show that the line with equation y = -3x + 6 is a tangent to the
circle with equation x2 + y2 + 10x - 2y - 14 = 0
Daily Practice
10.12.15
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.
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.
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 turning point
Q3. Write the the function y = 9x2 + 6x + 3 in completed square
form and state the turning point
1
S56 (5.3) Trig. Functions.notebook
December 17, 2015
Exact Values
Examples:
3. Find the exact value of sin
Today we will be continuing to learn about
x
0
π
cosx
1
­1
sinx
0
0
1
tanx
0
0
undefine
d
0
1
related angles.
Homework due Tuesday.
Daily Practice
11.12.15
Q1. The roots of the equation (x ‐ 1)(x + k) = ‐4 are equal. Find the
values of k.
Exact Values
Examples:
4. Find in its simplest form the exact value of
2 sin 210˚cos 330˚
Q3. Find the rate of change of 剹x when x = 1/8
Page 59
Ex. 4E
Q4. Find the equation of the altitude from B in the triangle A(7, 3),
B(1, 6)and C(‐5, ‐1)
x
0
π
cosx
1
­1
sinx
0
0
1
tanx
0
0
undefine
d
0
1
Solving Trig. Equations Algebraically
Note: If the question has no degree symbol above x, then the answer
is expected in radians.
Today we will be learning to solve trig.
equations.
HW due Tuesday.
Solve the equations like you would solve regular equations. Some
may even involve factorising.
S56 (5.3) Trig. Functions.notebook
Daily Practice
December 17, 2015
14.12.15
Today we will be solving trig. equations
HW due tomorrow
Solving Trig. Equations Algebraically
Solving Trig. Equations Algebraically
Examples:
1. Solve 3sinx0 + 2 = 0 where (00 ≤ x ≤ 3600)
Solving Trig. Equations Algebraically
Examples:
3. Solve 2cos3x0 - 1 = 0
Examples:
2. Solve 2sin2x0 - 1 = 0 where (00 ≤ x ≤ 3600)
Daily Practice
(From HSN notes)
15.12.15
S56 (5.3) Trig. Functions.notebook
December 17, 2015
Solving Trig. Equations Algebraically
1. Solve 2cos2x = 1 where (0 ≤ x ≤ 2π)
Today we will be continuing to learn how to solve trigonometric
equations.
Homework due
Solving Trig. Equations Algebraically
Examples: Using factorisation
(From Heinemann book)
2. 3sin2x0 - 4sinx0 + 1 = 0
Daily Practice
where (00 ≤ x ≤ 3600)
16.12.15
HW
Solving Trig. Equations with compound angles Algebraically
Compound angles are angles made up of more than one angle.
E.g. x0 + 450
Examples: Solve for x,
Today we will be learning to solve trig.
equations with compound angles.
1. cos (2x - 20)0 = 0.5
where (00 ≤ x ≤ 3600)
S56 (5.3) Trig. Functions.notebook
December 17, 2015
Solving Trig. Equations with compound angles Algebraically
2. 2cos(2x -
)=1
Daily Practice
where (0 ≤ x ≤ 2π)
17.12.15
Q1.
Today we will be continuing to solve trig. equations.