Term 3 : Unit 1 Trigonometric Functions

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Term 3 Unit 1Trigonometric Functions

• Name ____________ ( ) Class _____
  Date _____

1.1 Trigonometric Ratios and General Angles
1.2 Trigonometric Ratios of Any Angles
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Trigonometric Equations
1.1 Trigonometric Ratios and General Angles
Objectives
In this lesson, we will learn how to find the
trigonometric ratios for acute angles,
particularly those for 30, 45 and 60 (or
respectively in radians).
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Trigonometric Equations
Trigonometric Ratios of Acute Angles
The three trigonometric ratios are defined as
hypotenuse
hypotenuse
hypotenuse
opposite
opposite
opposite
adjacent
adjacent
adjacent
OPQ is a right angled triangle
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Trigonometric Equations
Example 1 In the right-angled triangle ABC, tan
? 2. Find sin ? and cos ?. Solution
2
Since tan ? ,
BC 2 units and AB 1 unit.
By Pythagoras Theorem, AC .
1
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Trigonometric Equations
Trigonometric Ratios of Special Angles
The length of the diagonal is v2 and the angle is
45.
Draw a diagonal to the square.
Draw a unit square.
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Trigonometric Equations
Trigonometric Ratios of Special Angles
The length of the altitude is v3 and the angles
are 60 and 30.
The altitude bisects the base of the triangle.
Draw an altitude.
Draw an equilateral triangle of side 2 cm.
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Trigonometric Equations
Trigonometric Ratios of Complementary Angles
In the right-angled triangle OPQ
but ?OPQ 90 ?
?OPQ ?
If ? is in radians
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Trigonometric Equations
Example 2 Using the right-angled triangle in the
diagram, show that sin(900 ?) cos ?. Hence,
deduce the value of
Solution
sin 700 sin (900 200)
cos 200
Thus, sin(900 ?) cos ?.
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Trigonometric Equations
Consider angles in the Cartesian plane.
OP is rotated in an anticlockwise direction
around the origin O. The basic (reference) angle
that OP makes with the positive xaxis is a.
Now OP is rotated in the clockwise direction.
2nd quadrant
1st quadrant
4th quadrant
3rd quadrant
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Trigonometric Equations
Example 3 Given that 00 lt ? lt 3600 and the basic
angle for ? is 400, find the value of ? if it
lies in the (a) 3rd quadrant, (b) 4th
quadrant. Solution (a) (b)
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Trigonometric Equations
Example 4
Using the complementary angle identity.
Substitute for sin ?.
Using the complementary angle identity.
Substitute for tan A.
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Trigonometric Equations
Example 5 Solution
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Trigonometric Equations
Example 6 Solution
Find all the angles between 0 and 360 which
make a basic angle of 70.
The angles are as follows
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Trigonometric Equations
1.2 Trigonometric Ratios of Any Angles
Objectives

• In this lesson, we will learn how to
• extend the definitions of sine, cosine and
  tangent to any angle,
• determine the sign of a trigonometric ratio of
  an angle in a
• quadrant,
• relate the trigonometric functions of any angle
  to that of its basic
• (reference) angle and solve simple
  trigonometric equations.

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Trigonometric Equations
The three trigonometric ratios are defined as
Trigonometric Ratios of Any Angles
y
y
y
r
r
r
x
x
x
PQ y
OQ x
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Trigonometric Equations
Example 7 Find the values of cos ?, sin ? and
tan ? when ? 1350. Solution
When ? 1350, 1800 ? 450.(basic angle)
P has coordinates (1, -1) and
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Trigonometric Equations
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Trigonometric Equations
Signs of Trigonometric Ratios in Quadrants
P has coordinates ( a, b )
P has coordinates ( a, b )
P has coordinates ( a, b ).
P has coordinates ( a, b )
? ( 360 a )
? ( 180 a )
? a
? ( 180 a )
1st quadrant
2nd quadrant
4th quadrant
3rd quadrant
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Trigonometric Equations
Signs of Trigonometric Ratios in Quadrants
For positive ratios
In the four quadrants
S (sin ?)
A ( all )
T (tan ?)
C (cos ?)
The signs are summarised in this diagram.
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Trigonometric Equations
Example 8 Without using a calculator, evaluate
cos 120. Solution
120 is in the 2nd quadrant, so cosine is negative
A
S
Basic angle,
T
C
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Trigonometric Equations
Basic Trigonometric Equations
Example 9 Find all the values of ? between 0 and
360 such that sin ? 0.5. Solution
Since sin ? lt 0, ? is in the 3rd or 4th quadrant,
For the basic angle,
A
S
T
C
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Trigonometric Equations
Example 10 Find all the values of ? between 0o
and 360o such that 2sin2 ? 1 0. Solution
? is in the 1st, 2nd, 3rd or 4th quadrant,
A
S
For the basic angle,
T
C