Mathematics

Key unit competence 

Use trigonometric circles and identities to determine trigonometric ratios and apply them to solve related problems. 

Learning objectives 

1.1 Trigonometric concepts 

The word trigonometry is derived from two Greek words: trigon, which means triangle, and metric, which means measure. So we can define trigonometry as measurement in triangles. 

Angle and its measurements 

Activity 1.1

In pairs, discuss what an angle is. Sketch different types of angles and name them: acute, obtuse, reflex, etc. Measure the angles to verify the sizes.

An angle is the opening that two straight lines form when they meet. In Figure 1.1, when the straight line FA meets the straight line EA, they form the angle we call angle FAE.  We may also call it “the angle at the point A,” or simply “angle A.” 

                        

The two straight lines that form an angle are called its sides.  And the size of the angle does not depend on the lengths of its sides. 

Degree measure 

To measure an angle in degrees, we imagine the circumference of a circle divided into 360 equal parts. We call each of those equal parts a “degree.” Its symbol is a small o:  1° = “1 degree.” 

                

The measure of an angle, then, will be as many degrees as its sides include.  To say that angle BAC is 30° means that its sides enclose 30 of those equal divisions. Arc BC is 30/180 of the entire circumference. 

Radians

Activity 1.2

This activity will be carried out in the field. Each group will need: two pointed sticks, two ropes (each about 50 cm in length), black board protractor and metre rule. 

1. Fix the pointed sticks, one at each end of the rope. 

2. Place the sharp end of one of the sticks onto the ground. 

3. With that point as the centre, let the tip of the 2nd stick draw a circle, radius 50 cm. 

4. Take the 2^nd rope also of length 50 cm and fit it on any part of the circumference (an arc). 

5. Take the metre rule and draw a line from each of the ends of the rope to the centre of the circle. 

6. Use the large protractor to measure the angle enclosed by the two lines. What do you get? 

7. Compare your result with the rest of the groups. Are they almost similar? 

8. Repeat the task using ropes of length 70 cm each. How do the results compare with the ones of 50 cm lengths?

The radian is a unit of angular measure. It is defined such that an angle of one radian subtended from the centre of a unit circle produces an arc with arc length of r. 

      

1 radian is about 57.2958 degrees. The radian is a pure measure based on the radius of the circle.

Degree-radian conversions

                           

There are π radians in a half circle and 180° in the half circle. 

So π radians = 180° and 1 radian  (approximately).

A full circle is therefore 2 radians. So there are 360° 

per 2 radians, equal to  or 57. 29577951°/ radian. Similarly, a right angle is  radians and a straight angle is   

Example 1.1 

Convert 45° to radians in terms of π. 

Solution

Example 1.2 

Convert  5π/6  to degrees

Solution

The unit circle 

Activity 1.3

Imagine a point on the edge of a wheel. As the wheel turns, how high is the point above the centre? In groups of four, represent this using a drawing.

The unit circle is the circle with centre (0, 0) and radius 1 unit.

          

Definition of sine and cosine

Activity 1.4

Work in groups. Consider a point P(x,y) which lies on the unit circle in the first quadrant. OP makes an angle q with the x-axis as shown in Figure 1.9.

The x is the side adjacent to the angle q. And y is the side opposite the angle q.The radius  of 1 unit is the hypotenuse.

Using right-angled triangle trigonometry: 

Note: We use cos² q for (cos q)² and sin² q for (sin q)².

Activity 1.5

In groups of three, use graph paper to draw a circle of radius 10 cm. Measure the half chord and the distance from the centre of the chord. Use various angles, say for multiples of 15°. Plot the graphs and determine the sines and the cosines. 

We can also use the quarter unit circle to get another ratio. This is tangent which is written as 'tan' in short.

Trigonometric ratios

Activity 1.6 

Carry out research on the trigonometric ratios. What are they? Define them.

There are three basic trigonometric ratios: sine, cosine, and tangent.  The other common trigonometric ratios are: secant, cosecant and cotangent. 

Trigonometric ratios in a right-angled triangle 

In Figure 1.12, the side H opposite the right angle is called the hypotenuse. Relative to the angle q, the side O opposite the angle q is called the opposite side. The remaining side A is called the adjacent side.

     

Trigonometric ratios provide relationships between the sides and angles of a right angle triangle. The three most commonly used ratios are sine, cosine and tangent. 

These six ratios define what are known as the trigonometric functions. They are independent of the unit used. 

The trigonometric ratios of the angles 30º, 45º and 60º are often used in mechanics and other branches of mathematics. So it is useful to calculate them and know their values by heart. 

The angle 45º

Activity 1.7

In pairs, draw an isosceles triangle where the two equal sides are 1 unit in length. Use Pythagoras' theorem to calculate the hypotenus.

In Figure 1.13, the triangle is isosceles. Hence the opposite side and adjacent sides are equal, say 1 unit. 

The hypotenuse is therefore of length  units (using Pythagoras’ theorem). 

  

The angles 60º and 30º 

Activity 1.8

In pairs, draw an equilateral triangle, ABC, of sides 2 units in length. Then draw a line AD from A perpendicular to BC.  AD bisects BC giving BD = CD = 1.

                                   

From this we can determine the following trigonometric ratios for the special angles 30º and 60º:

We can now complete the following table. 

Complementary angles Two angles are complementary if their sum is 

                 

These two angles (30° and 60°) of Figure 1.15 are complementary angles, because they add up to 90°. Notice that together they make a right angle 

Thus,   are complementary angles. 

Supplementary angles 

Two angles are supplementary if their sum is 180° (= π). The two angles (45° and 135°) of Figure 1.16 are supplementary angles because they add up to 180°.

            

Notice that together they make a straight angle. 

Thus, (π – q) and q are supplementary angles. 

For the two supplementary angles (π – q) and q, we have the following: 

Opposite angles 

Two angles are opposite if their sum is 0. Thus, – q and q are opposite angles. 

For the two opposite angles – q and q, we have the following: 

                          

Anti-complementary angles 

Two angles are anti-complementary if their difference is  

Notice that the two angles (60° and 150°) of Figure 1.18 are anti-complementary angles because their difference is 90°. 

Thus,   and q are anti-complementary angles. 

                  

For the two anti-complementary angles 

 

 Anti-supplementary angles

Two angles are anti-supplementary if their difference is 180° (= π).

Notice that the two angles (30° and 210°) of Figure 1.19 are anti-supplementary angles, because their difference is 180°. Sin (210°)  

Thus, (π + q) and q are anti-supplementary angles. 

         

For the two anti-supplementary angles (π + q) and q, we have the following: 

Coterminal angles 

Coterminal angles are angles in standard position (angles with the initial side on the positive x-axis) that have a common terminal side. For example 30°, –330° and 390° are all coterminal.

           

To find a positive and a negative coterminal angle with a given angle, you can add and subtract 360°, if the angle is measured in degrees, or 2π if the angle is measured in radians.

    

   

The trigonometric identities 

In mathematics, an identity is an equation which is always true. There are many trigonometric identities, but the one you are most likely to see and use is, 

Addition formulae 

1.2 Reduction to functions of positive acute angles Graphs of some trigonometric functions 

1.3 Triangles and applications 

Triangles 

The cosine rule 

For any triangle with sides a, b, c and angles ABC as shown in Figure 1.26, the cosine rule is applicable:

Note: For a right triangle at A i.e. A = 90°, cos A = 0. So a² = b² + c² – 2bc cos A; reduces to a² = b² + c² , and is the Pythagoras’Rule.

The cosine rule can be used to solve triangles if we are given:

    

Rearrangement of the original cosine rule formulae can be used to find angles if we know all three sides. The formulae for finding angles are: 








The sine rule

In any triangle ABC with sides a, b and c units in length, and opposite angles A, B and C respectively, 

Note: The sine rule is used to resolve problems involving triangles given either: 

   

Air navigation

Inclined plane 

Bearing