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Unit 3 Polynomial Functions, Equations and Inequalities

Chapter 6 Trigonometry 28

29 Chapter 6 Trigonometry

q

Chapter 6 Trigonometry

Name: _______________

6.1 Trigonometric Ratios for Any Angle in Standard PositionDate:

Standard Position: An angle is in standard position when the initial arm starts at (0, 0) and lies on the positive x-axis; the terminal arm is rotated about (0, 0).

Angles in standard position are graphed on the same plane as functions, and we label the four sections as quadrants:

=

=

=

=

=

=

=

=

=

side

opposite

side

tan

hypotenuse

side

cos

hypotenuse

opposite

side

sin

q

q

q

yx1234223223221234

yx

yx

Example 1: For each angle, draw the angle in standard position:

a)

=

120

q

b)

-

=

85

q

yx

yx

Suppose the ray continues to rotate. When the ray first reaches its starting position it forms an angle of ______ (if rotating counter-clockwise) and form an angle of ______ (if rotating clockwise). After rotating all around until a full cycle, you will end on the same position as standard position, this new angle will be either greater than 360 or less than 360.

Example 2: For each angle in example 1, determine the angle after a full cycle around.

When a ray rotates more than 1 revolution, it forms angles greater than 720 (when rotating counter-clockwise) and form angles less than 720 (when rotating clock-wise).

Example 3: For each angle in example 1, determine the angle after 2 full cycles around.

You can continuously adding or subtracting 360 and you will end up with the same terminal forever. These values are called coterminal angles.

Coterminal Angles: Two or more angles in standard position, when the position of P is the same for each angle. When represents any angle, then any angle coterminal with is represented by these expressions, where n is any integer.

n

360

q

, where

n

(natural number)

You TryIdentify the angles coterminal with 210 that satisfy the domain 720 720. Express the angles coterminal with 210 in general form.

Reference Angles

Definition: Consider the angle in standard position. The reference angle of is _____________________________________________________________________.

Sketch the reference angle, r, of each angle .

q

=

=

=

=

=

=

=

=

=

side

opposite

side

tan

hypotenuse

side

cos

hypotenuse

opposite

side

sin

q

q

q

yx551010510105

Unit Circle

The unit circle is simply a circle whose radius equals _______.

For our purposes, we will place the centre of a unit circle at the origin of the Cartesian plane.

Relating tan to sin and cos

Since

y

=

q

sin

and

x

=

q

cos

for the unit circle. Then

tan

=

=

x

y

q

Special Angles:

Recall the two special triangles:

Definition of Secant, Cosecant, and Cotangent

In mathematics, the ratios

q

cos

1

,

q

sin

1

,

q

tan

1

(reciprocal) are used so often that we give them special names.

We define the secant, cosecant, and cotangent functions are follows:

Secant of

Cosecant of

Cotangent of

Example 4: Determine the trigonometric ratio for each degree

in degrees

0

30

60

90

45

180

270

360

sin

cos

tan

sec

csc

cot

Example 5: Determine the exact value of the six trigonometric ratios of 495.

You Try Determine the exact value of the six trigonometric ratios of 150.

Determining Trigonometric Ratios of non-special triangles.

Example 6: Determine the exact value of the six trigonometric ratios of 200.

Determining All Trigonometric Ratios and Angle Measure given one Trigonometric Ratio.

Example 7: a) Suppose

13

12

cos

=

q

. Determine the exact values for the other trigonometric ratios for