Trigonometry & Circular Functions PDF

Foundation Mathematics Trig&CircFn TRIGONOMETRY & CIRCULAR FUNCTIONS INTRODUCTION The word “Trigonometry” comes from Greek words which mean “the measurement of triangles”. The trigonometric ratios (sine, cosine and tange...

Foundation Mathematics Trig&CircFn TRIGONOMETRY & CIRCULAR FUNCTIONS INTRODUCTION The word “Trigonometry” comes from Greek words which mean “the measurement of triangles”. The trigonometric ratios (sine, cosine and tangent) are defined for a right-angled triangle. The ratios are written as sin, cos and tan. opposite sin θ = hypotenuse hypotenuse adjacent cos θ = opposite hypotenuse opposite θ tan θ = adjacent adjacent MEASURING ANGLES When the line OA is rotated about O, the amount of turn is called an angle. Angles can be measured in degrees or radians. Degree measure y When the line OA is rotated through a full revolution, it A’ has been rotated through 360 degrees (written 360º) 1 Angle A'OA A degree is th of a full revolution 360 O A x Radian measure y Radian measure measures angles in terms of the radius of R a circle. 1 rad Definition : When an angle cuts off an arc on a circle R x equal to the radius of the circle, the amount of turn is 1 radian. 1 radian ≈ 57.3º Notes : 1 radian is written as 1c. If there is neither degree nor radian symbol, assume that the angle is in radians. The circumference of a circle is C = 2 π r. So for 1 full revolution, we turn through 2π radians. So 2π c = 360º ; which means : π c = 180 This can be used to convert any angle from degrees to radians, and vice versa. Page 1 of 21 Foundation Mathematics Trig&CircFn Example Convert to radians : (i) 135º (ii) 60º Solution πc 3π πc π (i) 135 = 135 × = (ii) 60  = 60  × = 180  4 180  3 Example 5π Convert to degrees : ( ii ) 2.5c ( i )             6 Solution 5π 5π 180 180 450 ( i )             = × = 150 ( ii ) 2.5c =2.5c × c = =143.2 ( to 4 significant figures) 6 6 π π π EXACT VALUES FOR TRIGONOMETRIC RATIOS 30º 45º 2 3 2 2 1 60º 60º 45º 1 1 1 θ sin θ cos θ tan θ 0 0 1 0 π 1 3 1 6 2 2 3 π 1 1 4 2 2 1 π 3 1 3 2 2 3 π ∞ 2 1 0 (undefined) Exercise 1 * When writing approximate answers, write to 4 significant figures 1. Express the following angles in radians: (a) 45º (b) 270º (c) 30º (d) 450º (e) 2 revolutions (f) 33º (g) 317º (h) 12.5º 2. Express the following angles in degrees: 2π c 7π c 5π 5π 7π (a ) (b) (c) (d) − (e) 3 6 2 12 4 ( f ) 2c ( g ) 1.5c ( h ) 4.21c ( i ) − 0.006c Page 2 of 21 Foundation Mathematics Trig&CircFn 3. Write the value of each of the following in exact form: πc πc π π ( ) a tan ( ) b cos ( c ) sin ( d ) tan ( e ) cos 0 3 6 2 2 ( f ) sin 45  ( g ) − cos 60 ( h ) tan 90   4. In each of the following find the ratio in exact form (in each case, θ is an acute angle): 3 ( a ) Find sin θ, given tan θ = 4 5 ( b ) Find cos θ, given tan θ = 12 4 ( c ) Find tan θ, given sin θ = 5 1 ( d ) Find cos θ, given tan θ = 2 TRIGONOMETRIC FUNCTIONS OF ANGLES The trigonometric ratios can also be defined in terms of a unit circle (radius 1 unit). These are called trigonometric functions A ray is drawn from the origin at an angle θ from the positive x-axis. It meets the unit circle y at P. The triangle OPQ has a hypotenuse of 1 unit. (0,1) A tangent line is drawn to the circle at (1,0) P(θ) d 1 sin θ = y y θ cos θ = x (–1,0) x (1,0) x sin θ tan θ = =d cos θ (0, –1) THE FOUR QUADRANTS The unit circle can be divided into four quadrants as y shown at right. π 1st quadrant : 0 < θ < 2 2nd 1st π π–θ θ 2nd quadrant :