Trigonometry Review Tutorial Shorter Version

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1 Author: Michael Migdail-Smith Originally developed: 007 Last updated: June 4, 0 Tutorial Shorter Version Avery Point Academic Center

2 Trigonometric Functions The unit circle. Radians vs. Degrees Computing Trig Ratios Trig Identities Functions Definitions Effects Applications Michael Migdail-Smith, University of Connecticut, 007

3 Review hypotenuse adjacent Starting with a right triangle, like the one pictured on the right, three basic trig ratios are defined as follows: opposite sin cos opposite hypotenuse adjacent hypotenuse opposite tan adjacent sin cos Michael Migdail-Smith, University of Connecticut, 007

4 Review hypotenuse adjacent Three additional trig ratios are defined from the basic ratios as follows: opposite sec cos csc sin hypotenuse adjacent hypotenuse opposite cot tan adjacent opposite Michael Migdail-Smith, University of Connecticut, 007 Table of Contents 4

5 The Unit Circle Consider the unit circle: a circle with a radius equal to one unit, centered at the origin. The unit circle has a circumference: C the unit circle. radians deg rees Michael Migdail-Smith, University of Connecticut, Distance around the unit circle is measured in radians. Radians relate directly to degrees: The distance around the unit circle, starting at the point (, 0) equals the angle formed between the -ais and the radius drawn from the origin to a point along 60 5

6 The Unit Circle Radians vs. Degrees The conversion from radians to degrees or the other way around uses the equation: radians deg rees 80 Convert 0 to radians by solving the equation: 0 80 Cross multiply to solve for : Michael Migdail-Smith, University of Connecticut, 007 6

7 The Unit Circle Radians vs. Degrees The conversion from radians to degrees or the other way around uses the equation: radians deg rees 80 Convert Practice 5 4 radians to degrees by solving the equation: Cross multiply to solve for : Michael Migdail-Smith, University of Connecticut,

8 The Unit Circle Computing Trig Ratios The trigonometric ratios can be computed using the unit circle. To form the trig ratios, we need a right triangle inscribed in the unit circle, with one verte placed at the origin so that the perpendicular sides are parallel to the -ais & y-ais. y This triangle has the following relationships: hypotenuse = = cos y = sin tan = y/ Notice that tan is the same as the slope of the line radiating out of the origin! Michael Migdail-Smith, University of Connecticut, 007 8

9 = cos y = sin tan = y/ / / The Unit Circle Computing Trig Ratios Using the newly defined relationship, the trig ratios are determined by reading the & y values off the graph. 0 Note the pattern: Values increase from 0 to according to integral square roots. Michael Migdail-Smith, University of Connecticut, 007 angle sine

10 The Unit Circle Computing Trig Ratios These trig ratios are summarized in the following table: sin cos tan /6 /4 / / 0 Michael Migdail-Smith, University of Connecticut, 007 Table of Contents 0

11 Trig identities In the first and forth quadrants is positive while y changes sign. As is swept up and down away from the positive -ais, only its sign changes. These characteristics lead to the following relationships: cos (-) = cos () sin (-) = -sin () tan (-) = -tan () Michael Migdail-Smith, University of Connecticut, 007

12 Trig identities y From the first to the second quadrants changes sign while y remains positive. As is swept up away from the positive and negative -ais, equal angle sweeps are related as: : -. These characteristics lead to the following relationships: cos (-) = -cos () sin (-) = sin () tan (-) = -tan () Michael Migdail-Smith, University of Connecticut, 007

13 Trig identities - Eamples: a.) second quadrant: 5 sin sin 6 6 b.) fourth quadrant: tan 7 tan 4 4 c.) third quadrant: 7 cos cos 6 sin 6 Michael Migdail-Smith, University of Connecticut, 007 tan 4 cos 6 6 cos cos 6 6 Practice

14 Trig identities Combining the Pythagorean Theorem with the properties of the right triangle inscribe in the unit circle we get the following trig identity, relating sine to cosine: sin + cos = y (cos,sin ) Note that when = sin, cos (,0) Michael Migdail-Smith, University of Connecticut, 007 4

15 Trig identities A similar triangle combined with the Pythagorean Theorem produces the trig identity relating tangents to secants: sec = + tan sec (,tan ) Cosecant and Cotangent are similarly related: csc = + cot (,0) Michael Migdail-Smith, University of Connecticut, 007 5

16 Trig identities These other trig identities can also be derived from the unit circle: cos(-) = coscos + sinsin cos(+) = coscos - sinsin cos() = cos - sin sin(+) = sincos + cossin sin(-) = sincos - cossin These trig identities are useful to solve problems such as: cos cos cos cos sin sin Michael Migdail-Smith, University of Connecticut, 007 Proof Table of Contents 6

17 Functions Consider the ratio epressed as a function: y sin : f sin We can graph the function on the Cartesian coordinates: y y Michael Migdail-Smith, University of Connecticut, 007 7

18 Functions - Definition The function: f sin has the domain:, and range:, y Michael Migdail-Smith, University of Connecticut, 007 8

19 Functions - Definition The function: f cos has the domain:, and range:, y Michael Migdail-Smith, University of Connecticut, 007 9

20 Functions - Definition The function: f tan has the domain: 0,,,... and range:, 4 y Michael Migdail-Smith, University of Connecticut, 007 0

21 Functions - Effects y = Asin (B-C)+D Amplitude (A): Distance between minimum and maimum values. Frequency (B): Number of intervals required for one complete cycle Period (/B): Length of interval containing one complete cycle Phase Shift (C): Shift along horizontal ais. Vertical Shift (D): Shift along vertical ais. Michael Migdail-Smith, University of Connecticut, 007

22 y = A(sin (B-C) Eamples: Functions - Amplitude (A) y sin y y sin Michael Migdail-Smith, University of Connecticut, 007

23 Functions Frequency/Period (B) y = A(sin (B-C) Eamples: y sin y Period = / y sin Period = Michael Migdail-Smith, University of Connecticut, 007

24 y = A(sin (B-C) Eamples: y y sin Functions Phase (C) sin cos y - Practice Michael Migdail-Smith, University of Connecticut, 007 4

25 Functions - Applications What does the sine curve represent? Periodic Behavior: Sound Waves, Tides Springs Cyclic growth and decay Consider the waves in the ocean, The amplitude effect their height Choppy water is caused a high frequency Flat seas indicate that there is a low frequency and amplitude Michael Migdail-Smith, University of Connecticut, 007 5

26 Functions - Applications Low tide occurs in some port at 0:00 am on Monday and again at 0:4 pm that same night. At low tide the water level is foot and at high tide it measures 7 feet. What is the sine function that represents the water level? f t 4 sin0. 507t Amplitude: The difference between low and high tide is 7-=6 feet. The amplitude is half that difference: 6/= feet Vertical Shift: 7 The average water level: 4 ft. Frequency: Time between high tides: hrs. 4 min. =.4 hrs. Period : Michael Migdail-Smith, University of Connecticut, 007 Table of Contents Practice 6

27 Practice:. Epress 5 in radians: Convert 4/ radians to degrees: Return Michael Migdail-Smith, University of Connecticut, 007 7

28 Practice: Epress the following trig ratios as multiples of a simple radical epression: sin sin sin tan 4 tan 4 tan 4 cos cos Michael Migdail-Smith, University of Connecticut, 007 Continued 8

29 Michael Migdail-Smith, University of Connecticut, Epress the following trig ratios as multiples of a simple radical epression: Return Practice: sin tan 4 cos sin sin 4 cos 4 cos tan tan tan

30 A. B. Practice: Match the curve to the equation: y sin 4 y sin 4 y B C. y sin 4 A C - Return Michael Migdail-Smith, University of Connecticut, 007 0

Math 1205 Trigonometry Review

Math 1205 Trigonometry Review Math 105 Trigonometry Review We begin with the unit circle. The definition of a unit circle is: x + y =1 where the center is (0, 0) and the radius is 1. An angle of 1 radian is an angle at the center of