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Topic 5: Trigonometry
 

 

1. INTRODUCTION

Trigonometry (from Greek trigonõn=triangle and metron=measure) is the study of how the sides and angles of a triangle are related to each other.

2. MEASURE OF ANGLES

The angles are measured in three units which are:

- degree (D),
- grades (G), and
- radian (R)

Once we have the angle measured in one of the above units, we can convert it in other units using the following relation:

Where D stands for degree, R for radians, G for grades and

This relation can be split into 3 relations:

Example

Convert to radians and grades

Solution

           

Thus,

Example

Convert 20 grades to radians and degrees

Solution

                     

Thus, 20 grades = 18 deg. Also, 20 grades = 0.314 radians

3. UNIT CIRLCE

A unit circle is a circle with a radius of one. Frequently, especially in trigonometry, the unit circle is the circle of radius one centered at the origin (0,0) in the Cartesian coordinate system in the Euclidean plane.

If (x, y) is a point on the unit circle, then |x| and |y| are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean Theorem, x and y satisfy the equation .

The unit circle is divided into 4 parts called quadrants. Each quadrant measures 90 degrees, means that the entire circle measures 360 degrees or radians.

The interior of the unit circle is called the open unit disk, while the interior of the unit circle combined with the unit circle itself is called the closed unit disk.

4. TRIGONOMETRIC RELATION

Consider the following circle with radius r

  

Observing the triangle in the above circle, we see that r is the hypotenuse, x is the adjacent side and y is the opposite side. Then,

For unit cirle, the radius is 1 and hence,

Other relations

Hence,

5. TRIGONOMETRIC NUMBER OF REMARKABLE ANGLES

Rule for constructing the above table

- Write down the angles 0 degrees, 30 degrees, 45 degrees, 60 degrees and 90 degrees in ascending order.
- To find sine for each angle,
             * In the row of sine rank each angle starting by 0
             *Find the square roots of each rank
             *Divide the result by 2.
- To find cosine, reverse the result obtained in row of sine
- To find tangent, divide sine by cosine

Other remarkable angles

Consider the following unit trigonometric circle

From above figure we have

                                           

Thus the table trigonometric number of remarkable angle is

Remark

Sine is positive in 1st and 2nd quadrants and negative in 3rd and 4th quadrants while Cosine is positive in 1st and 4th quadrants and negative in 2nd and 3rd quadrants.

6. REDUCTION TO THE FIRST QUADRANT

6.1. EQUIVALENT ANGLES

Two angles are equivalent if their difference is . This means that the angle are equivalents.

Example

If are equivalent we write . From example above we may write . This means that they are on the same point on trigonometric circle.

The equivalent angles have the same trigonometric numbers. Then

6.2. Negative angles

The angle β is negative of the angle if . In this case we write

For negative angles the following identities are true

6.3. Complementary angles

Two angles are said to be complementary if their sum is 90 deg. are complementary.

Example

The angles 40 deg and 50 deg are complementary angles since their sum is 90 deg.

For two complementary angles the following identities are true

6.4. Supplementary angles

Two angles are said to be supplementary if their sum is 180 deg.

Example

The angels 100 deg and 80 deg are supplementary angles since theur sum is 180 deg

For two supplementary angles the following identities are true

7. TRIGONOMETRIC FORMULAE

Remember that

7.1. Addition formulae

Remember that

                                       

The basic addition formula is

Replace y by -y in relation (a), we have

Replace x by in relation (a), we have

Replace y by -y in relation (c), we have

Divide (b) by (c), we have

Divide the numerator and denominator of right hand side by , we have

Replace y by -y in relation (e), we have

Divide members of relation (c) by the members of relation (b), we have

Divide the numerator and denominator of right hand side by , we have

Replace y by -y in relation (g), we have

Hence, the addition formulae are

Addition formulae are useful when finding trigonometric number of some angles.

Example

Use addition formulae and table of remarkable angles to find

Solution

7.2. Derived formulae

From relation (1), replace y by x, we have

This relation is called the fundamental relation of trigonometry.

From this relation we can write

From relation (2), replace y by x, we have

Combining relations (12) and (10), we have

Combining relations (12) and (11), we have

From relation (3), replace y by x, we have

From relation (5), replace y by x, we have

From relation (7), replace y by x, we have

From relation (14), we can write

Divide the right hand side by , we have

Divide numerator and denominator of the right hand side by , we have

From relation (15), we can write

Divide the right hand side by , we have

Divide numerator and denominator of the right hand side by , we have

Example

 

7.3. Transformation of sum in product and vice versa

From relation (1), (2), (3), and (4) (1) + (2) gives

(1) - (2) gives

(3) + (4) gives

(3) - (4) gives

Hence formulae for transforming product in sum are

In above formulae for transformation of products in sum, let

Now,

Then, the formulae for transforming sum in product are

Example

 

8. TRIGONOMETRIC EQUATIONS AND INEQUALITIES

When solving trigonometric equation or inequality try to transform or rearrange the given expressions using trigonometric identities to remain with a simple equation or inequality. Simple equation or inequality involves one trigonometric function with one unknown, like .

We need first to know the following relations:

 or

Where the superscript -1 means the inverse of a function. We will see more on inverse functions later.

Also remember the following identities:

                                      

8.1. Trigonometric equations

As we saw it on unit trigonometric circle, sine or cosine of any angle cannot be greater than 1 as it cannot be less than -1. If you find the case where sine or cosine falls out of the interval [-1,1], in that case there is no solution.

Example

Solve in set of real numbers the equation

Solution

Here we need to know the angle whose sine is . Using table of remarkable angle or scientific calculator we find that , so that . This is not the only solution since we know that .

So is another solution.

Also . Then,

In general we can write

8.2. Trigonometric inequalities

When solving inequalities, first replace the inequality sign by equal sign and then solve. Find the two no equivalent angles in . Place these two angles on a trigonometric circle. The two points divide the circle into two arcs. Choose the arcs containing the angles corresponding to the given inequality.

Example

Solve the inequality

Solution

Since we are given the condition  then the solution set is

If the condition was not given, the solution set is

9. APPLICATIONS

9.1. Cosine and sine laws

Cosine law

Cosine law (also known as cosine formula or cosine rule) relates the lengths of sides to the cosine of one of the angles.

Consider the following triangle

The cosine law says that

Example

How long is the side c in the following figure?

Solution

The formula says

Sine law

The sine law (or sine formula or sine rule) is an equation relating the lengths of the sides of a triangle to the sine of its angles. If are a, b, c the lengths of the sides of a triangle and A, B, C are the opposite angles respectively, then the sine law is

   or 

Example

Calculate side c

Solution

9.2. Bearings

We say that point B has a bearing of θ degrees from point A if the line connecting A to B makes an angle of θ with a vertical line drawn through A, the angle being measured clockwise.

Example

Jim and Ted live on one side of the river, and Martha lives on the other side. The distance across the river is 100 yards. Ted, who lives downstream from Matha, measures an angle of 35 degrees between the shoreline and a straight line leading to Martha's house. Jim, who lives upstream from Martha, measures an angle of 60 degrees. How far apart do Ted and Jim live?

Solution

The relative positions of Martha, Jim, and Ted are represented in this picture:

Example

A private plane flies for 1.3 hours at 110 mph on a bearing of 40°. Then it turns and continues another 1.5 hours at the same speed, but on a bearing of 130°. At the end of this time, how far is the plane from its starting point?

Solution

The bearings tell us the angles from "due north" in a clockwise direction. Since 130 – 40 = 90, the two bearings give us a right triangle. From the times and rates, we have

Now, let's give the geometrical shape to our problem and set up a triangle:

Using the Pythagorean theorem, we get

Hence the plane is approximately 218 miles away at the end of the time.

9.3. Angle of elevation and angle of depression

If the object is below the level of the observer, then the angle between the horizontal and the observer's line of sight is called the angle of depression.

The angle of elevation of an object as seen by an observer is the angle between the horizontal and the line from the object to the observer's eye (also known as the line of sight).

Example

Andrew was flying a kite on a hill, but he dumped his kite into the pond below. If the length of the string of his kite is 150 meters and the angle of depression from his position to the kite is , then how high is the hill where he is standing?

Solution

Let's first draw a diagram for a better understanding of the problem:

Now,

Hence the hill is 75 meters above the lake.

Example

The angle of elevation of the top of an incomplete vertical pillar at a horizontal distance of 100 meters from its base is 45 degrees. If the angle of elevation of the top of the complete pillar from the same spot is to be 60 degrees, then by how much the height of the incomplete pillar should be increased?

Solution

Let's draw a diagram to figure out the situation:

In triangle ABC, we know that

Similarly, in triangle, we know that

And

Hence the height of the incomplete pillar is to be increased by to complete the pillar

 

 
 

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