θ = 45 θ = 135 θ = 225 θ = 675 θ = 45 θ = 135 θ = 225 θ = 675 Trigonometry (A): Trigonometry Ratios You will learn:

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1 Trigonometr (A): Trigonometr Ratios You will learn: () Concept of Basic Angles () how to form simple trigonometr ratios in all 4 quadrants () how to find the eact values of trigonometr ratios for special angles (,, 4, 6, 9, 8, ) (4) how to evaluate the values of trigonometr ratios of an magnitude, in degrees. () The concept of Basic Angles Consider angles in a Cartesian plane: The and aes divide the plane into the st, nd, rd and 4 th quadrants in an anticlockwise direction as shown. An initial line segment A on the positive ais is rotated in the anticlockwise direction until it reaches. The angle θ is measured from the positive -ais and it is said to be in the quadrant where lies. All angles are measured from the positive -ais. Angles measure in a anti-clockwise sense is taken to be positive. Angles measure in a clockwise sense is taken to be negative. For first quadrant, the angle is between o and 9 o. For second quadrant, the angle is between 9 o and 8 o. For third quadrant, the angle is between 8 o and 7 o. For fourth quadrant, the angle is between 7 o and 6 o. The angle is the first quadrant is known as the basic angle/reference angle. (i) Representing Angles in Cartesian lane θ 4 θ θ θ 67 θ 4 θ θ θ 67

2 (ii) Trigonometric Ratios of An Angles st Quadrant nd Quadrant th Quadrant 4 th Quadrant θ sin θ cosθ tanθ o 4 o 6 o bservations: S (sin) A (all) T (tan) C (cos) ACRNYM: Let us define trigonometric ratios for an angle θ (acute,, and obtuse, refle etc) in terms of the coordinates ( ) (,) the length, r, where r +. sinθ r cosθ r tanθ r θ

3 First Quadrant Coordinates of ( ),, A sin cos tan A - Second Quadrant sin cos (, ), angle α Coordinates of tan Third Quadrant Coordinates of,, angle α ( ) sin A - - cos tan A α - Fourth Quadrant sin cos tan (, ), angle α Coordinates of (iii) Negative Angles sin ( θ ) cos( θ ) tan ( θ )

4 Eample Find, without using a calculator, the value of each of the following (a) sin ( º) (b) cos ( º) (c) tan ( º) (iv) Basic Trigonometric Equations Basic trigonometric equations are of the form sinθ k, cosθ k and tanθ k where k is a constant. The steps to solve such equations are:. Consider the sign of k to identif the quadrants in which θ lies.. Find the basic angle,α, where α is acute.. Find all the values of θ in the required interval. Eamples Given that 4º. cos 6, find, without using calculators, the values of sin 4º, cos 4º and tan Given that tan 4, find, without using calculators, the values of sin º, cos º and tan º. If A is an acute angle and sin A 4, find, without using calculators, the value of tan A and of cos A. 4 If B is an obtuse angle and cos B, find, without using calculators, the value of sin B and of tan B. Given that ta n C m, and 8 C 7, find, without using calculators, the value of si n C and of cosc. 6 Given that cosθ 4, and 8 o < θ <, evaluate 7 o tanθ and sinθ. 7 Given that tanθ and that tan θ and sinθ have opposite signs, find the value of cosθ and of sinθ. 8 Given that tan A and that tan A and cos A have opposite sign, find the value of cos A and o of cos(9 A). 4

5 9 Given 6 <A<7 and that A has a basic angle of 8, find all the values of A if A is in the nd Quadrant. Given that sin β p where β is an acute angle, obtain an epression, in terms of p, for (a) tan β, (b) cos β, (c) sin ( 8 β ) (d) cos( 8 β ) +,, (e) tan ( 6 + β ), (f) tan ( 6 β ) o o Find all angles, where < < 6 such that (a) sin (b) tan (c) sin 7. (d) 7 cos 4( cos ) (e) (sin )(sin + ) (f) 8sin sin + Eercise: Q Without using a calculator, determine whether these trigo ratios are positive or negative. (a) sin (b) cos 4 (c) tan (d) cos (e) tan 4 (f) sin 6 (g) cos ( 6 ) (h) tan ( ) Q For each of the following conditions, determine the quadrant(s) that A must lie. (a) tan A > (b) cos A > and sin A < (c) cos A and tan A are of the same sign (d) sin A and tan A are of opposite signs Q Given that sin A, and 9 <A<8, find the value of cos A and and tan A. Q4 Given that cos X, tan Y 4 and that X and Y are in the same quadrant, evaluate without the use of calculator, tan X cosy.

6 Q Find all the angles between 6 and 8 such that (a) sin A (b) cos A Q6 Solve the following equation for 8. (i) sin.67 (ii) tan (iii) tan cos 4 Q7 Solve for 8 8, tan Q8 Find all angles between and 6 inclusive such that (a) cos.7 (b) tan.7 (c) sin.866 (d) tan (e) cos (f) 4(tan ) ( tan ) (g) sin + tan 7 8 cos + (h) cos Q9 Find the values of where <<6, such that (a) (sin )(sin + ) (b) (cos )(cos + ) (c) sin ( cos ) (d) tan ( cos ) (e) ( sin )(tan + ) (f) sin ( + ).7 (g) cos ( ).4 (h) tan ( 78 ).7 (i) sin (. 7 ).6 (j) cos (74 ).4 (k) sin ( + ) + tan 7 Q Find all angles, where 8 <<8, that satisf the equation (a) cos + 4 (b) tan ( + ) 6 7sin + 6 (c) sin (d) cos 4 (e) cos cos + 6

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