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Review of Trigonometric Functions

Trigonometric Identities [Note that sin2θ is used to represent (sinθ)2.]
Pythagorean Identities:   Reduction Formulas:  
Sum or Difference of Two Angles:   Half–Angle Formulas:   Double–Angle Formulas:
Law of Cosines:   Reciprocal Identities:   Quotient Identities:


  Evaluating Trigonometric Functions

There are two ways to evaluate trigonometric functions: (1) decimal approximations
with a calculator (or a table of trigonometric values) and (2) exact evaluations using
trigonometric identities and formulas from geometry. When using a calculator to
evaluate a trigonometric function, remember to set the calculator to the appropriate
mode—degree mode or radian mode.

EXAMPLE 2 Exact Evaluation of Trigonometric Functions


The angle π/3 in standard position
Figure D.32


Evaluate the sine, cosine, and tangent of π/3.

Solution Begin by drawing the angle in standard position, as shown in
Figure D.32. Then, because radians, you can draw an equilateral triangle
with sides of length 1 and θ as one of its angles. Because the altitude of this triangle
bisects its base, you know that Using the Pythagorean Theorem, you obtain

Now, knowing the values of x, y, and r, you can write the following.

NOTE All angles in this text are measured in radians unless stated otherwise. For example,
when we write sin 3, we mean the sine of 3 radians, and when we write we mean the sine
of 3 degrees.

The degree and radian measures of several common angles are given in the table
below, along with the corresponding values of the sine, cosine, and tangent (see
Figure D.33).

Common First Quadrant Angles

The quadrant signs for the sine, cosine, and tangent functions are shown in Figure
D.34. To extend the use of the table on the preceding page to angles in quadrants other
than the first quadrant, you can use the concept of a reference angle (see Figure
D.35), with the appropriate quadrant sign. For instance, the reference angle for 3π/4
is π/4 and because the sine is positive in the second quadrant, you can write

Similarly, because the reference angle for 330° is  30° and the tangent is negative in
the fourth quadrant, you can write

Figure D.35

Common angles
Figure D.33


Quadrant signs for trigonometric functions
Figure D.34



  EXAMPLE 3 Trigonometric Identities and Calculators

Evaluate the trigonometric expression.


a. Using the reduction formula you can write

b. Using the reciprocal identity you can write

c. Using a calculator, you can obtain

Remember that 1.2 is given in radian measure. Consequently, your calculator must
be set in radian mode.

Solving Trigonometric Equations

How would you solve the equation ? You know that is one solution,
but this is not the only solution. Any one of the following values of θ is also a

You can write this infinite solution set as is { nπ :n an integer }


  EXAMPLE 4 Solving a Trigonometric Equation

Solution points of
Figure D.36

Solve the equation

Solution To solve the equation, you should consider that the sine is negative in
Quadrants III and IV and that

Thus, you are seeking values of θ in the third and fourth quadrants that have a reference
angle of π/3In the interval ,the two angles fitting these criteria are


By adding integer multiples of 2π to each of these solutions, you obtain the following
general solution.

,where n is an integer.

(See Figure D.36.)