Basics of Trigonometry

1.Define the trigonometric ratios using sinθ, cos θ and tan θ, using right angles triangles. 2.Extend the definitions for sinθ, cos θ and tan θ for 0°< θ<360° 3.Define the reciprocals of trigonometric ratios cosec θ, sec θ, cot θ 4.Derive values of trigonometric ratios for special cases (without using a calculator) θЄ{0 °; 30 °; 45 °; 60 °; 90 °} 5.Solve two dimensional problems involving right angled triangles. 6.Solve simple trigonometric equations for acute angles. 7.Use diagrams to determine the numerical values of ratios for angles from 0 ° to 360 °.

1. Investigate trigonometric ratios in right – angled triangles. In the given right angled triangle: PQR PR is the hypotenuse PQ is the opposite side of angle R QR is the adjacent side to angle R The three ratios for this angle become: R Q P y x r

1. Investigate trigonometric ratios in right – angled triangles. R Q P This format of the definition of trigonometry ratios are used to solve triangles y x r

Reciprocals of Trig Ratios R Q P This format of the definition of trigonometry ratios are used to solve triangles y x r

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SPECIAL TRIANGLES ° ° 60° √3 √2

Test your knowledge Use your knowledge of special triangles to solve the following: a) cos 45°b) sin 30° c) tan 60 °d) sin 45° + cos 30° e) 4cos²30°+ tan30°.sin60°

2. Solve right – angled triangles by finding angles and sides Three elements of the triangle must be given of which one must be a side. To solve a triangle we need to determine the sizes of the sides and the angles as well as determining the area of the triangle. If two sides are given, the length of the third side is determined by using the theorem of Pythagaros. A B C 25

2. Solve right – angled triangles by finding angles and sides Worked Example: Determine the length of AC and AB A B C 25

Example 2 In the figure: TS = 26,5m ; TQ = 12,8m Determine : Reason: Sum of the angles of a triangle 12, T Q S Angle of elevation Angle of depression

3. Construct and interpret geometric models True Bearings: The direction is measured from north. Measured in a clockwise direction Are written as a three digit number e.g. S E N W S E N W

3. Construct and interpret geometric models Compass bearings means the direction is measured from either North or South e.g. It is measured as an acute angle in either a clockwise or anti – clockwise direction towards the direction under discussion. It can be written as: S E N W S E N W

4. Construct and interpret trigonometric models A quantity surveyor’s work involves a lot of measuring so that highways can be planned properly. Consider a model of a hill ABCD, 50m high and rising at an angle of 18º with respect to the horizontal. DC = 400m and ABCD is a rectangle.

4. Construct and interpret trigonometric models

Solution or:

Investigate the effect of on the co – ordinates of the point P(x;y) An angle is in standard position if the vertex coincides with the origin on a Cartesian plane ( the X and Y axes system). Rotation always starts at the positive OX - axis. If the rotation is anti – clockwise, positive angles are formed and if it is clockwise, negative angles are formed. Y X - P(x;y) r y x 1 st Quad2 nd Quad 3 rd Quad 4 th Quad y 0

Investigate the effect of on the co – ordinates of the point P(x;y) As increases from,its value affects the coordinates of P(x;y). P can lie in any of the four quadrants, depending on the values of x and y The x - coordinate changes from r to 0 in the 1st quadrant From 0 to -r in the second quadrant From –r to 0 in the 3rd quadrant and again from 0 to r in the 4th quadrant Y X - P(x;y) r y x 1 st Quad2 nd Quad 3 rd Quad 4 th Quad y 0

Test Your Knowledge 1. Determine the height of a pole (RS) if the shadow of the pole (TS) is 8m and the angle of elevation of the sun is Answer A) 5,67m B) 7,32m C) 5,32mD) 4,62m R S T 8m

The y – coordinate changes from: 0 to r in the first quadrant r to 0 in the second quadrant 0 to –r in the second quadrant and from -r to 0 in the fourth quadrant. If OP = r, then: and P(+;+) r > 0 P(-;+) r > 0 P(-;-) r > 0 P(+;-) r > 0 y x

Investigate the effect of the co – ordinates of P(x;y) on For every value of we can define six trigonometry ratios and our studies are only based on three of these:

Investigate the effect of the co – ordinates of P(x;y) on These three form the basis of Trigonometry and because of the properties of a fraction, the following are important: a fraction = 0, if the numerator =0 a fraction is undefined, if the denominator = 0 a fraction is positive if both numerator and denominator have the same signs (+/ both - ) a fraction is negative if the numerator and denominator have opposite signs. As the values of becomes bigger in the first quadrant, the values of the co-ordinates changes.

Investigate how affects the ratios: If r is constant As the value of increases over the interval from, the value of sine increases from 0 to 1. At the same time the value of the cosine function decreases from 1 to 0. The tangent of an angle grows from 0 to infinity, because is undefined The sine function increases if the angle increases in this interval y x0

Investigate how r affects the ratio if is constant Note : The angle remained the same, but the radius (r) and the co – ordinates of the point P change. E.g. y y y xxx

Test Your Knowledge If: Calculate the sine, cosine and tangent for in using the above sketch. You may leave any irrational answers in surd form r x y

Solutions

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