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Math 4 S. Parker Spring 2013 Trig Foundations

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The Trig You Should Already Know Three Functions: Sine Cosine Tangent

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The Trig You Should Already Know Definitions: Sine = opp/hyp Cosine = adj/hyp Tangent = opp/adj

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The Trig You Should Already Know All the trig you have studied so far has been based upon the sides of a ________ triangle. right

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The Trig You Should Already Know So far you have used trig to find: missing sides using sin /cos /tan missing angles using sin -1 / cos -1 / tan -1

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The Trig You Will Learn You will find that trig functions can be defined: by the sides of a right triangle (prior knowledge) based upon other trig functions based upon the unit circle

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The Trig You Will Learn There are six (6) trig functions: Sine Cosine Tangent Cosecant Secant Cotangent The three you already know

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The Three Reciprocal Definitions

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Given One Trig Function, Find Others Write definitions of given and needed functions. Use Pythagorean Theorem to find missing side. adjacent hypotenuse opposite x˚

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Angles in Standard Position Vertex is always at the origin. Initial side is always on the positive x axis. Terminal side is the ending side.

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Angles in Standard Position Positive angle = counterclockwise 0˚ 90˚ 180˚ 270˚

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Angles in Standard Position Negative angle = clockwise 0˚ −270˚ −180˚ −90˚

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Angles in Standard Position Quadrantal angle = angle not in a quadrant: 0˚, 90˚, 180˚, 270˚, 360˚, etc. Quadrantal angles will not use reference angles.

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Coterminal Angles Coterminal angles always differ by a multiple of 360. Every angle has an infinite number of coterminal angles. The interval given determines how many and which coterminal angles may be used.

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Reference Angles All reference angles are acute. An acute angle does not need a reference angle (or is considered its own reference). Quadrantal angles NEVER use reference angles.

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Reference Angles Finding Reference Angles: 1 st Quadrant: No ref. angle 2 nd Quadrant: 180 − angle 3 rd Quadrant: angle − 180 4 th Quadrant: 360 − angle

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Reference Angles for Angles > 360˚ If the given angle is greater than 360˚, first find a coterminal that falls in the interval 0˚≤ x < 360˚. Now find the reference angle based upon the coterminal angle.

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Reference Angles for Angles < 0˚ If the given angle is negative, first find a coterminal that falls in the interval 0˚≤ x < 360˚. Now find the reference angle based upon the coterminal angle. Remember: What is true about ALL reference angles?

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Radians and Degrees

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Common Degrees and Radians As the semester goes along, we will use degrees and radians interchangeably.

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Trig and the Unit Circle

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Tangent

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Cotangent (cot)

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Cosecant (csc)

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Secant (sec)

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Trig With Reference Angles If angle given is not acute, first find the reference angle. Consider whether the trig function is positive or negative in this quadrant. Find answer based upon showing these two pieces of information.

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Trig With Reference Angles

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Point on the Terminal Side

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The hypotenuse will always be the missing side. Pay attention to quadrant to decide whether answer is positive or negative. Use trig definitions to find answer.

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θ hypotenuse adjacent opposite There are 6 trig ratios that can be formed from the acute angle θ. Sine θ= sin θCosecant θ= csc θ Cosine θ= cos θSecant.

θ hypotenuse adjacent opposite There are 6 trig ratios that can be formed from the acute angle θ. Sine θ= sin θCosecant θ= csc θ Cosine θ= cos θSecant.

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