UNIT 2 : Set IR of real numbers►

Unit 1: Fundamentals of Trigonometry

  • Unit 1: Fundamentals of Trigonometry

    My goals

    Introduction
    Trigonometry studies relationship involving lengths and angle of a triangle.
    The techniques in trigonometry are used for finding relevance in navigation particularly satellite systems and
    astronomy, naval and aviation industries, oceanography, land surveying, and in cartography (creation of maps).
    Now those are the scientific applications of the concepts in trigonometry, but most of the mathematics we study would seem (on the surface) to have little real-life application. Trigonometry is really relevant in our day to day activities. The following pictures show us the areas where this science finds use in our daily activities.
    In this unit we will see how we can use this to resolve problems we might encounter.



                Subsidiary Mathematics for Rwanda Secondary Schools Learner’s Book 4

    Required outcomes
    After completing this unit, the learners should be able to:
    »» Define sine, cosine, and tangent (cosecant, secant and
    cotangent) of any angle – know special values.
    »» Convert radians to degree and vice versa.
    »» Use trigonometric identities.
    »» Apply trigonometric formulae in real world problems.

    1. Trigonometric concepts


    Trigonometry is the study of how the sides and angles of a triangle are related to each other. A rotation angle is formed by rotating an initial side through an angle, about a fixed point called vertex, to terminal position called terminal side.
    It is drawn in what is called standard position if the initial side is on the positive x-axis and the vertex of the angle is at the origin.




    Angles in standard position that have a common terminal side are called co-terminal angles; the measure of smallest positive rotation angle is called principal angle. Angle is positive if rotated in a counterclockwise direction and negative when rotated clockwise. Angles are named according to where their terminal side lies, for instance, the x-axis and y-axis divide a plane into four quadrants as follow


    Example 2
    Draw each of the following angles in standard position and which of these angles is co-terminal to 300 ?
    a) 300 b) 1200 c) -2300 d) 7500 e) -3300

    Subsidiary Mathematics for Rwanda Secondary Schools Learner’s Book 4





    Example 3
    Draw each of the following angles in standard position and indicate in which quadrant the terminal side is.

























    Subsidiary Mathematics for Rwanda Secondary Schools Learner’s Book 4

    Subsidiary Mathematics for Rwanda Secondary Schools Learner’s Book 4




    Notice: Unit circle



         

    A unit circle is a circle of radius
    one centered at the origin (0,0) in
    the Cartesian coordinate system
    in the Euclidean plane.
    In the unit circle, the coordinate
    axes delimit four





    quadrants that are numbered in an anticlockwise direction. Each quadrant measures 90 degrees, meaning that the entire circle measures 360 degrees or 2π radians.

    Exercise 2


    Trigonometric ratios of acute angles
    Activity 3
    Construct two right angled triangles, one of which is an enlargement of the other.

    How is the side opposite to the right angle (or the longest side)
    called?
    For both triangles, consider an angle and compute the following
    ratios.
    • Opposite side to the considered angle and hypotenuse.
    • Adjacent side and hypotenuse.
    • Opposite side to the considered angle and adjacent side.
    How can you conclude?

    Consider the following circle with radius r.

    In this triangle, we define the following
    six ratios:
    • The three primary trigonometric values
    . The ratio x/r is called cosine of the angle
    α , noted cos α .

    • The ratio y/r is called sine of the angle α ,
    noted sin α 

    • The ratio y/x is called tangent of the angle
      α , noted tan α .












    Example 10
    Calculate the six trigonometric values for the diagram.









    Solution
    Use adjascent = 3,opposite = 4 and hypotenuse = 5


    Example 11



    For each angle, calculate
    the reciprocal trigonometric
    values.








    Example 12




    Exercise 3
    For each angle, calculate the reciprocal trigonometric values


    Trigonometric Number of special Angles 300, 450, 600











    From the figure, we have;


    CAST Rule

    Activity 5
    Copy and complete the table to summarise the signs of the trigonometric values in each quadrant.



    The following diagram shows which primary trigonometric values are positive in each quadrant. This is called the CAST rule.



    Sine is positive in first and second quadrant but negative in third and fourth quadrant. Cosine is postive in the first and fouth quadrant but negative in second and third quadrant Tangent is negative in in the first and third quadrants but negative in second and fourth quadrant.







    Trigonometric identities

    Activity 6
    Here is a right triangle.




    Basic Rules





    Exercise 6


    2. Triangle and applications
    Solving triangle
    Solving a triangle is to find the length of its sides and measures of its angles. There are two methods for solving a triangle: cosine law and sine law.

    Cosine law








    Exercise 7




    Sine law



    CX is perpendicular to side AB and BY is perpendicular to side AC. Let AB =c, AC=b, BC=a, CX=h and BY=k
    1. In triangle BCX find sin B. In triangle AXC find sin A. Deduce the relationship between side a and side c.
    2. In triangle ABY find sin A. In triangle BCY find sine C.Deduce the relationship between side b and side a.
    3. Deduce relationship between three sides.
    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 a,b,c are the lengths of the sides of a triangle and A, B,C are the opposite angles respectively, then the

    sine law is




    Exercise 8



    Applications
    Many real situations involve right triangles. Using angles and trigonometric functions, we can solve problems
    involving right triangle. We have already seen how to solve a triangle.







    If B is east of A then the bearing is 90o.








    Similarly, if B is south of A then the bearing is 180o, and if B is west of A then the bearing is 270o. The bearing can be any number between 0 and 360, because there are 360 degrees in a circle. We can also use right triangles to find distances using angles given as bearings. In navigation, a bearing is the direction from one object to another. Further, angles in navigation and surveying may also be given in terms of north, east, south, and west. For example, N70  E refers to an angle from the north, towards
    the east, while N70 0W W refers to an angle from the north, towards the west.
























    Example 19

    A ship travels on a N500E course. The ship travels until it is due north of a port which is 10 kilometers due east of the port from which the ship originated. How far did the ship travel? 

    solution

    Example 20

    An airplane flies on a course of S300E  , for 150 km. How far south is the plane from where it originated? 

    Solution 

    Using known information, consider the following figure:

    2. Angle of elevation and angle of depression 

    You can use right triangles to find distances, if you know an angle of elevation or an angle of depression. The figure below shows each of these kinds of angles. 

    Suppose that an observer is standing at the top of a building and looking straight ahead at the birds (horizontal line). The observer must lower his/her eyes to see the car parked (slanting line). The angle formed between the two lines is called the angle of depression.


    Suppose that an observer is standing at the top of a building and looking straight ahead at the birds (horizontal line). The observer must raise his/her eyes to see the airplane (slanting line).  The angle formed between the two lines is called the angle of elevation.

    Example 21 

    The angle of elevation of the top of a pole measures 48 from a point on the ground 18 metres away from its base. Find the height of the flagpole. 

    Solution 



    Example 22 

    An airplane is flying at a height of 2 kilometres above the level ground. The angle of depression from the plane to the foot of a tree is 15o. Find the distance that the air plane must fly to be directly above the tree.

    Solution

    Let x be the distance the airplane must fly to be directly above the tree.  The level ground and the horizontal are parallel, so the alternate interior angles are equal in measure. 

    So, the airplane must fly about 7.46 kilometres to be directly above the tree. 

    3. Inclined plane 

    An inclined plane, also known as a ramp, is a flat supporting surface tilted at an angle, with one end higher than the other, used as an aid for raising or lowering a load. On the inclined plane the weight of the object causes the object to push into and, the object slides, to rub against the surface of the incline. Also the weight causes the object to be pulled down the slant of the incline. The component that pushes the object into the surface is 

    Example 23 

    An object with a mass of 2.5 kg is placed on an inclined plane. The angle of the inclined is 20 degrees. What are the parallel force and the perpendicular force? (g=9.8m/s2) 

    Solution



    Unit summary 

    1. Trigonometry is the study of how the sides and angles of a triangle are related to each other. A rotation angle is formed by rotating an initial side through an angle, about a fixed point called vertex, to terminal position called terminal side. Angle is positive if rotated in a counterclockwise direction and negative when rotated clockwise. 

    2. The amount we rotate the angle is called the measure of the angle and is measured in: degree, grade or radian. 

    4. Unit circle is a circle of radius one centered at the origin (0,0) in the Cartesian coordinate system in the Euclidian plane. In the unit circle, the coordinate axes delimit four quadrants that are numbered in an anticlockwise direction. 

    Each quadrant measures 90 degrees, means that the entire circle measures 360 degrees or   radians.

    9. Applications 

    Many real situations involve right triangle. Using angles and trigonometric functions, we can solve problems involving right triangle like: 

    • Bearings and air navigation 

    • Angles of elevation and angle of depression 

    • Inclined pla

UNIT 2 : Set IR of real numbers►