Mathematics for Computer Graphics. Trigonometry

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1 Mathematics for Computer Graphics Trigonometry

2 Trigonometry...????? The word trigonometry is derived from the ancient Greek language and means measurement of triangles. trigonon triangle + metron measure = Trigonometry

3 Trigonometry... is all about Triangles a C b B A c

4 Opposite Right Angled Triangle A right-angled triangle (the right angle is shown by the little box in the corner) has names for each side: Adjacent is adjacent to the angle "θ Opposite is opposite the angle θ Adjacent The longest side is the Hypotenuse.

5 DEGREE MEASURE AND RADIAN MEASURE B O 1 1 B 1 1 A O Initial Side A Degree measure: derives from defining one complete rotation as 360 o 1 degree (1 ) is the angle for a rotation from the initial side to terminal side that is (1/360) th of a revolution. Degree measure = 180/ π x Radian measure 1 degree =? Radian Radian measure: the angle created by a circular arc whose length is equal to the circle s radius. 1 radian = 1 unit of arc length in a circle of radius 1 unit. Radian measure= π/180 x Degree measure 1 Radian =? degree

6 ANGLES Angles (such as the angle "θ" ) can be in Degrees or Radians. Here are some examples: Angle Degree Radians Right Angle 90 π/2 Straight Angle 180 π Full Rotation 360 2π

7 Trigonometric Ratio..

8 Opposite "Sine, Cosine and Tangent" The three most common functions in trigonometry are Sine, Cosine and Tangent. They are simply one side of a triangle divided by another or known as trigonometric ratios. For any angle "θ": Sine Function: sin(θ) = Opposite / Hypotenuse Cosine Function: cos(θ) = Adjacent / Hypotenuse Tangent Function: tan(θ) = Opposite / Adjacent θ Adjacent

9 Their graphs as functions and the characteristics

10 TRIGONOMETRIC FUNCTIONS 0 π/6 π/4 π/3 π/2 π 3π/2 2π sin 0 1/2 1/ 2 3/ cos 1 3/2 1/ 2 1/ tan 0 1/ Not defined 0 Not defined 0

11 Opposite Other Functions (Cotangent, Secant, Cosecant) Similar to Sine, Cosine and Tangent, there are three other trigonometric functions which are made by dividing one side by another: Cosecant Function : csc(θ) = Hypotenuse / Opposite Secant Function : sec(θ) = Hypotenuse / Adjacent Cotangent Function : cot(θ) = Adjacent / Opposite θ Adjacent

12 Computing unknown sides or angles in a right triangle. To find a side of a right triangle you can use the Pythagoras Theorem, which is a 2 +b 2 =c 2. The a and b represent the two shorter sides and the c represents the longest side which is the hypotenuse To find unknown angles and side you can use the trigonometric ratios α Example Given a triangle where the hypotenuse = 10 and one angle, = Find the other sides (a and b).

13 Inverse Trigonometric Ratios The sin, cos and tan functions convert angles into ratios, and the inverse functions sin 1, cos 1 and tan 1 convert ratios into angles. For example, sin(45 o ) = 0.707, therefore sin 1 (0.707) = 45 o. Although sine and cosine functions are cyclic functions (i.e. they repeat indefinitely) the inverse functions return angles over a specific period sin -1 (x) = θ where π/2 θ π/2 and sin(θ) = x cos -1 (x) = θ where 0 θ π and cos(θ) = x tan -1 (x) = θ where π/2 θ π/2 and tan(θ) = x

14 TRIGONOMETRIC RELATIONSHIPS

15 Between sin, cos, tan cos(β) = sin(90 o β) sin(β) = cos(90 o β) cos(β) = sin(β + 90 o ) sin(β) / cos(β) = tan(β) Between sin, cos, tan and csc, sec, cot csc(β)= 1/sin(β) sec(β) = 1/cos(β) cot(β) = 1/tan(β)

16 Pythagorean Identities h y α x x 2 + y 2 = h 2 x 2 + y 2 = 1 h 2 h 2 y 2 + x 2 = 1 h h sin 2 (α) + cos 2 (α) = 1

17 Pythagorean Identities h y x cos 2 (θ) + sin 2 (θ) = 1 cos 2 (θ) + sin 2 (θ) =. 1. cos 2 (θ) cos 2 (θ) cos 2 (θ) 1 + tan 2 (θ) = sec 2 (θ)

18 The Sin Rule B c a A b C a /sin(a) = b /sin(b) = c /sin(c) Example: A = 50 o, B = 30 o, a = 10, find b. Solution: b/sin(30) = 10 /sin(50) b = 10 sin(30)/ sin(50) b =

19 The Cosine Rule c A B a a 2 = b 2 + c 2 2bc cos(a) b 2 = c 2 + a 2 2ca cos(b) c 2 = a 2 + b 2 2ab cos(c) a = b cos(c) + c cos(b) b = c cos(a) + a cos(c) c = a cos(b) + b cos(a) b C

20 Compound Angles Two sets of compound trigonometric relationships show how to add and subtract two different angles The most common relationships sin(α + β) = sin(α) cos(β) + cos(α) sin(β) sin(α β) = sin(α) cos(β) cos(α) sin(β) cos(α + β) = cos(α) cos(β) sin(α) sin(β) cos(α β) = cos(α) cos(β) + sin(α) sin(β) tan(α + β) = tan(α) + tan(β) 1 tan(α) tan(β) tan(α β) = tan(α) tan(β) 1 + tan(α) tan(β)

21 Compound Angles Using the previous relationships, we can show the relationships for multiples of the same angle. sin(2α) = 2 sin(α) cos(α) cos(2α) = cos 2 (α) sin 2 (α) cos(2α) = 2 cos 2 (α) 1 cos(2α) = 1 2 sin 2 (α) sin(3α) = 3 sin(α) 4 sin 3 (α) cos(3α) = 4 cos 3 (α) 3 cos 2 (α) sin 2 (α) = 1/ 2 (1 cos(2β)) cos 2 (α) = 1 /2 (1 + cos(2β))

22 Exercise If A= 120 0, c=10, b= 8, Find a, B, and C. Solution: a 2 = b 2 + c 2 2bc cos(a) a 2 = (8)(10)cos(120) a 2 = (160)(-0.5) = = 244 a = a/sin(a) = c/sin(c) sin(c) = c(sin(a))/a sin(c) = (10)(sin(120))/15.62 sin(c) = (10(0.8660))/15.62 = C = sin -1 (0.5544) = B = = B c a A b C

MTH 112: Elementary Functions

MTH 112: Elementary Functions 6.2: Right triangle trigonometry 1/16 Figure: Euclid of Alexandria was a Greek mathematician, often referred to as the Father of Geometry 6.2: Right triangle trigonometry 2/16 6.2: Right triangle trigonometry