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Trigonometry Refresher 0011 0010 1010 1101 0001 0100 1011 By Erin Hill [email protected] 1 2 4 What it’s used for: 0011 0010 1010 1101 0001 0100 1011 • • • • • Trigonometry is best applied to: Geometry Astronomy Physics Advanced mathematics Almost any engineering field 1 2 4 How far to your friend’s house from your house? 0011 0010 1010 1101 0001 0100 1011 Friend’s House ? a Your House b Grocery Store 1 2 4 Triangles and Pythagorean Theorem: 0011 0010 1010 1101 0001 0100 1011 Pythagorean theorem: c 2 a 2 b2 This rule works only when using a right triangle. (Special case of the law of cosines). 1 c a 2 b2 a b 2 4 How far to your friend’s house from your house? 0011 0010 1010 1101 0001 0100 1011 Friend’s House 1 ? Your House θ b Grocery Store 2 4 Triangles and Pythagorean Theorem: 0011 0010 1010 1101 0001 0100 1011 Only know the length of one side and the value of one angle. c=? a=? 1 θ 2 4 b Use trigonometry to solve for either of the unknown sides (again only for a right triangle) Unit Circle • “b” = length of the 0011 0010 1010 1101 0001 0100 1011 right triangle on the x-axis (x-value) • “a” = height of the right triangle on the y-axis (y-value) • “c” = distance between the origin and our point (b,a). (c = 1 for unit circle). y (b,a) 1 c θ 1 b a 2 4 x Trig Functions: SOHCAHTOA y 0011 0010 1010 1101 0001 0100 1011 sin(θ) = opp/hyp = a/c cos(θ) = adj/hyp = b/c c=1 1 θ b = cos(θ) tan(θ) = opp/adj = a/b a = opposite side to angle, θ. b = adjacent side to angle, θ. c = hypotenuse side to angle, θ. tan(θ) = sin(θ)/cos(θ) 2 a = sin(θ) 4 x Similar Triangles 0011 0010 1010 1101 0001 0100 1011 tan(θ) = a/b tan(θ) = c/d a/b = c/d c a θ b d 1 2 4 Same angle: ratio of sides must be the same! Angles in degrees: • There are 360 degrees in a circle. y 0011 0010 1010 1101 0001 0100 1011 90 60 45 135 30 • Some of the common angles are shown to the right. • All positive angles start from the xaxis. 1 180 225, -135 2 0, 360 x 4 315, -45 Angles in radians (standard unit): • There are 2 radians in a circle. y 0011 0010 1010 1101 0001 0100 1011 /2 /3 /4 3/4 • Some of the common angles are shown to the right. /6 1 • All angles start from the x-axis. 5/4 2 4 0, 2 x Angles in radians (standard unit): y 0011 0010 1010 1101 0001 0100 1011 • 2 = 360 = 180 • Calculator has radian and degree mode, so check before you calculate! /2 /3 /4 3/4 /6 1 5/4 2 4 0, 2 x Convert these: 0011 0010 1010 1101 0001 0100 1011 From degrees to radians: From radians to degrees: • 25 • /7 • -140 • 4 1 2 4 Convert these: 0011 0010 1010 1101 0001 0100 1011 From degrees to radians: From radians to degrees: • 25 /180 = 5/36 • /7 180/ 25.7 1 2 4 • -140 /180 = -7/9 • 4 180/ 229.2 • or: -140 + 360 = 220 220 /180 = 11/9 Try these: 0011 0010 1010 1101 0001 0100 1011 c=? a=? 30 3 1 2 4 Try these: 0011 0010 1010 1101 0001 0100 1011 cos(30) = 3/c c*cos(30) = 3 c = 3/cos(30) = 3.46 c = 3.46 1 2 a = 1.73 tan(30) = a/3 a = 3tan(30) = 1.73 30 3 4 All Star Trig Class y 0011 1010 1101 and 0001tangent 0100 1011 All:0010 sine, cosine, all (0,1) have positive values in this quadrant. Star: only sine has positive values in this quadrant. II Star Trig III 1 All (-1,0) Trig: only tangent has positive values in this quadrant. Class: only cosine has positive values in this quadrant. I 2 4 (0,-1) Class IV x (1,0) Calculating Angle Use Inverse functions to “undo” the trig functions: 0011 0010 1010 1101 0001 0100 1011 • sin(θ) = 0.5 • cos(θ) = 0.5 • tan(θ) = 0.5 sin-1(c) = arcsin(c) = asin(c) • sin-1(sin(θ))= sin-1(0.5) θ = sin-1(0.5) = 30 1 2 • cos-1(cos(θ)) = cos-1(0.5) θ = cos-1(0.5) = 60 4 • tan-1(tan(θ)) = tan-1(0.5) θ = tan-1(0.5) = 26.57 Trig Functions: Periodic θ =0010 0 cos(θ) 1 and sin(θ) 0011 1010 1101=0001 0100 1011= y 0: point on circle is on x-axis. (0,1) θ = 90 cos(θ) = 0 and sin(θ) = 1: point on circle is on y-axis. θ = 180 cos(θ) = -1 and sin(θ) = 0: point on circle is on (-x)-axis. θ = 270 cos(θ) = 0 and sin(θ) = -1: point on circle is on (-y)-axis. 1 c=1 θ (-1,0) (b,a) 2 a = sin(θ) 4 b = cos(θ) x (1,0) (0,-1) Sine and cosine increase and decrease as you go around the circle Trig Functions: Periodic 0011 0010 1010 1101 0001 0100 1011 sin(θ) cos(θ) 1 1 0.5 0.5 θ 0 0 0 pi/2 pi 3pi/2 0 2pi -0.5 -0.5 -1 -1 pi/2 pi 1 3pi/2 2 4 θ 2pi y-axis Shift 0011 0010 1010 1101 0001 0100 1011 3 + cos(t) General form: A + Bcos(ωt + φ) 4 3.5 1 3 2.5 • A: Shifts the function up (+) or down (-) along the y-axis. 2 1.5 1 0.5 0 -0.5 0 pi/2 -1 Dotted line represents the original cos(t) function. 2 4 pi 3pi/2 2pi t Amplitude 0011 0010 1010 1101 0001 0100 1011 0.5cos(t) General form: A + Bcos(ωt + φ) • B (Amplitude): Shrinks (small #) or stretches (large #) along the yaxis. 1 0.5 0 0 -0.5 -1 pi/2 pi 1 3pi/2 2 4 2pi t Frequency 0011 0010 1010 1101 0001 0100 1011 General form: A + Bcos(ωt + φ) • ω (frequency): Shrinks (large #) or stretches (small #) along the xaxis. (1/s = Hz) • T (period) = 2/ω: Time it takes for the wave to complete one full cycle. (s) cos(0.5t) 1 0.5 0 0 -0.5 -1 pi/2 pi 1 3pi/2 2 4 2pi t Phase Shift 0011 0010 1010 1101 0001 0100 1011 General form: A + Bcos(ωt + φ) • φ: Shifts right (-) or left (+) along the x-axis. cos(t+pi/4) 1 0.5 0 0 -0.5 -1 pi/2 pi 1 3pi/2 2 4 2pi t If you shift cosine by φ = -/2, your function becomes sine! Common Trig Mistakes 0011 0010 1010 1101 0001 0100 1011 • θ = x/sin 1 • sin(90) = 0.894 2 4 Why are these wrong? sin(θ) 1 0011 0010 1010 1101 0001 0100 1011 0.5 θ 0 0 -0.5 -1 pi/2 pi 3pi/2 1 2pi 2 4 0011 0010 1010 1101 0001 0100 1011 b a c • sin(θ) = b/c 1 2 4 What’s wrong with this? • sin(θ2) = sin2(θ) and 0011 0010 1010 1101 0001 0100 1011 • sin-1(θ) = 1/sin(θ) 1 2 4 What’s wrong with this picture? Trigonometric Identities y 0011 0010 1010 1101 you 0001have 0100 a 1011 More commonly, (0,r) circle of radius, r. x2 + y2 = r2 r2cos2(θ) + r2sin2(θ) (x,y) = 1 r r2 θ (-r,0) x = rcos(θ) cos2(θ) + sin2(θ) = 1 divide by cos2(θ): 1 + sin2(θ)/cos2(θ): = 1/cos2(θ) 1 + tan2(θ): = sec2(θ) (0,-r) 2 y = rsin(θ) x (r,0) 4 sec(θ) = 1/cos(θ), csc(θ) = 1/sin(θ), cot(θ) = 1/tan(θ) For more trig help… 0011 0010 1010 1101 0001 0100 1011 Come to the QSC! UW2-030 www.uwb.edu/qsc 425.352.3170 [email protected] 1 2 4