Appendix D: Trigonometry Review

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Appendix D Trigonometry Review
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A triangle in which one angle is a right angle is
called a right triangle. The side opposite the
right angle is called the hypotenuse, and the
remaining two sides are called the legs of the
triangle.
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Acute Angle
c
Right Triangle
b
a
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Six Trigonometric Ratios
Formal Name Abbreviation Ratio
Cosine of ? cos ?
Sine of ? sin ?
Tangent of ? tan ?
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Six Trigonometric Ratios
Formal Name Abbreviation Ratio
Secant of ? sec ?
Cosecant of ? csc ?
Cotangent of ? cot ?
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Example Find the value of each of the six
trigonometric functions of the angle ?.
c Hypotenuse 13
a Adjacent 12
a2 b2 c2
122 b2 132
b2 132 - 122
b 5
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a Adjacent 12 b Opposite 5 c Hypotenuse
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Reciprocal Identities
Quotient Identities
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The inverse (or arc) trigonometric functions will
return an angle for a given real number.
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Use your calculator to evaluate the following.
Explain the relationship between the two sets of
numbers.
(d) Why doesnt this relationship hold for
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Trigonometric Functions
We know that angles (either in degrees or
radians) are essentially real numbers (any number
you can think of). Therefore, the trigonometric
ratios can be thought of as functions where the
input is the angle (any real number). f(x) cos
x
Notes If a degree symbol appears on the angle
measurement, it is given in degrees. If nothing
appears after the angle measurement, it is given
in radians. Be sure to set your calculator to the
appropriate mode.
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Coordinates of Points
Determine a and b as functions of x.
P(a, b)
r
b
x
a
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Trigonometric Functions

• The triangle formed by dropping a perpendicular
  line from the terminal side of an angle to the
  closest side of the x-axis is called a reference
  triangle.
• We will not use x and y to reference the point P.
  The variable x is reserved angle measurements.
• It is very important to include the sign with the
  values of a and b when labeling your reference
  triangle. This will assure that the values of the
  six trigonometric ratios are correct.

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Pythagorean Identities
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Other Identities

• Many more trigonometric identities can be found
  on page A29 of your text. These identities may be
  used in this and further calculus classes.

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Graphs of Trigonometric Functions
f(x) cos x Domain (-8, 8) Range -1, 1
f(x) sin x Domain (-8, 8) Range -1, 1
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Graphs of Trigonometric Functions
f(x) tan x Domain (-8, 8) Range -1, 1