**TOPIC 5: TRIGONOMETRY ~ MATHEMATICS FORM 4**

**Trigonometric Ratios**

The Sine, Cosine and Tangent of an Angle Measured in the Clockwise and Anticlockwise Directions

Determine the sine, cosine and tangent of an angle measured in the clockwise and anticlockwise directions

The basic three trigonometrical ratios are sine, cosine and tangent which are written in short as Sin, Cos, and tan respectively.

Consider the following right angled triangle.

Also we can define the above triangle ratios by using a unit Circle centered at the origin.

If θis an obtuse angle (90

^{0}<θ<180^{0}) then the trigonometrical ratios are the same as the trigonometrical ratio of 180^{0}-θIf θis a reflex angle (180

^{0}< θ<270^{0}) then the trigonometrical ratios are the same as that of θ- 180^{0}If θis a reflex angle (270

^{0}< θ< 360^{0}), then the trigonometrical ratios are the same as that of 360^{0 }-θWe

have seen that trigonometrical ratios are positive or negative

depending on the size of the angle and the quadrant in which it is

found.

have seen that trigonometrical ratios are positive or negative

depending on the size of the angle and the quadrant in which it is

found.

The result can be summarized by using the following diagram.

Trigonometric Ratios to Solve Problems in Daily Life

Apply trigonometric ratios to solve problems in daily life

Example 1

Write the signs of the following ratios

- Sin 170
^{0} - Cos 240
^{0} - Tan 310
^{0} - sin 30
^{0}

a)Sin 170

^{0}Since 170

^{0}is in the second quadrant, then Sin 170^{0 }= Sin (180^{0}-170^{0}) = Sin 10^{0}∴Sin 170

^{0}= Sin 10^{0}b) Cos 240

^{0}= -Cos (240^{0}-180^{0)}= -Cos 60^{0}Therefore Cos 240

^{0}= -Cos 60^{0}c) Tan 310

^{0 }= -Tan (360^{0}-310^{0}) = – Tan 50^{0}Therefore Tan 310

^{0}= -Tan 50^{0}d) Sin 300

^{0}= -sin (360^{0}-300^{0}) = -sin 60^{0}Therefore sin 300

^{0}= – Sin 60^{0}**Relationship between Trigonometrical ratios**

The above relationship shows that the Sine of angle is equal to the cosine of its complement.

Also from the triangle ABC above

Again using the ΔABC

b

^{2}= a^{2}+c^{2}(Pythagoras theorem)And

Example 2

Given thatA is an acute angle and Cos A= 0.8, find

- Sin A
- tan A.

Example 3

If A and B are complementary angles,

*Solution*If A and B are complementary angle

Then Sin A = Cos B and Sin B = Cos A

Example 4

Given that θand βare acute angles such that θ+ β= 90

^{0}and Sinθ= 0.6, find tanβ

*Solution*Exercise 1

For practice

**Compound Angles**

The Compound of Angle Formulae or Sine, Cosine and Tangent in Solving Trigonometric Problems

Apply the compound of angle formulae or sine, cosine and tangent in solving trigonometric problems

The aim is to express Sin (α±β) and Cos (α±β) in terms of Sinα, Sinβ, Cosαand Cosβ

Consider the following diagram:

From the figure above <BAD=αand <ABC=βthus<BCD=α+β

From ΔBCD

For

Cos(α±β) Consider the following unit circle with points P and Q on it

such that OP,makes angleα with positive x-axis and OQ makes angle βwith

positive x-axes.

Cos(α±β) Consider the following unit circle with points P and Q on it

such that OP,makes angleα with positive x-axis and OQ makes angle βwith

positive x-axes.

From the figure above the distance d is given by

In general

Example 15

1. Withoutusing tables find the value of each of the following:

- Sin 75°
- Cos105

Example 16

Exercise 4

1. Withoutusing tables, find:

- Sin15°
- Cos 120°

2. FindSin 225° from (180°+45°)

3. <!–[endif]–>Verify that

- Sin90° = 1 by using the fact that 90°=45°+45°
- Cos90°=0 by using the fact that 90°=30°+60°

4. <!–[endif]–>Express each of the following in terms of sine, cosine and tangent of acute angles.

- Sin107°
- Cos300°

5. <!–[endif]–>By using the formula for Sin (A-B), show that Sin (90°-C)=Cos C

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