Section 3.2 Unit Circles with Sine and Cosine

SECTION 3.2 UNIT CIRCLES WITHSINE AND COSINE

I) REVIEW:

SOH-CAH-TOA

Pythagorean Theorem:

Circle Equation:

Note: The coordinate of

any point on the circumference

will satisfy the circle equation

Ratio

Function

REVIEW: SPECIAL TRIANGLES

There are two types of special triangles:

30°, 60°, 90° triangle (Equilateral Triangle) and

45°, 45°, 90° triangle (Isosceles-Right Triangle)

Cut it in half

Equilateral Triangle

All sides and angles are equal

Use Pythagorean Thm. to findthe hypotenuse of the triangle

Isosceles-Right Triangle

Two sides are equal and one angle

Is equal to 90°

Use Pythagorean Thm. to findthe height of the triangle

Isoceles  2 equal angles

Special triangles can be used to find the exact value of sine/cosine/tangent of basic angles like: 30°, 45°, 60°, and 90°

Rather than obtaining a decimalrepresentation, we get the exact value as a fraction using special triangles

Ex: Use the special triangles to determine the exact valuefor each of the following:

II) UNIT CIRCLES:

In a unit circle, the radius (terminal arm) is 1 unit long

The tip of the terminal arm is given by the coordinates (x,y)

the terminal arm can be used to make a right triangle, so thesum of both the “x” & “y” coordinates squared will be equal to 1

Using the Pythagorean Thm:

Any point on the circumference of a unit circle will satisfy the equation!

III) SINE & COSINE IN UNIT CIRCLES:

When given the angle in std. position, we can find the coordinates of any point on the circumferenceusing trigonometry

The coordinates of any point on the circumference of aunit circle can be represented by:

Y-coordinate

X-coordinate

Note: r = 1

Ex: Given the angle in standard position in a unit circle, findthe coordinates of each of the following points P(x,y).

IV) SINE/COSINE/TANGENT IN DIFFERENT QUADRANTS

The sine/cosine/tangent of angles in each of the four quadrants willeither be positive or negative

Use the reference angle to determine whether if the ratio is either +’veor –’ve

sin/cos/tan of all angles in the 1st Quadrant will be positive

Only sine of angles in the 2nd Quadrant will be positive

Cosine & Tangent in the 2nd Quad will be negative

Only Tangent of angles in the 3rd Quadrant will be positive

Sine & Cosine in the 3rd Quad will be negative

Only Cosine of angles in the 4th Quadrant will be positive

Sine & Tangent in the 4th Quad will be negative

The previous table can be used to determine whichquadrant an angle will be in when given the ratio of a trigfunction

Ex: Given each of the following trig. functions and its ratios,determine which quadrants the angle can be in:

The ratio is negative

The ratio is positive

The ratio is negative

Ex: Solve for the angle, given the following trig functions.

The ratio is positive

The angle has tobe in Q1 and Q2

Inverse sine the

equation to find

where the angle is

Find the 2nd angle in Q2 using

the reference angle

Note: When given a trig equation, there

are usually 2 answers

Check:

Practice: Solve for the angle:

The ratio is negative

The angle has tobe in Q2 and Q3

Inverse cosine the

equation to find

where the angle is

Find the 2nd angle in Q3 using

the reference angle

Note: When given a trig equation, there

are usually 2 answers

Check:

V) USING SPECIAL TRIANGLES TO SOLVE TRIG EQUATIONS

For angles with reference angles of 30°, 45°, 60°, we can usespecial triangles

Draw the angle in standard position

Find the reference angle

Draw the 30°, 60°, 90° triangle

Draw the angle in standard position

Find the reference angle

Draw the 30°, 60°, 90° triangle

Draw the angle in standard position

Find the reference angle

Draw the 45°, 45°, 90° triangle

Draw the angle in standard position

Find the reference angle

Draw the 30°, 60°, 90° triangle

Draw the angle in standard position

Find the reference angle

Draw the 45°, 45°, 90° triangle

VI) FINDING THE COORDINATES OF POINT P ON THETERMINAL ARM

When given the ratio of a trig function, you will be asked to findthe coordinates of the endpoint on the terminal arm

Method #1) Find the value of the central angle

Using central angle, we can find the coordinates of point P byusing:

Note: There are usually two central angles, use reference angles

Method #2) Finding the exact value using the Pythagorean Thm.

Create a right triangle in the corresponding quadrants

Use the ratios of the sides to find sinθ and cosθ

Section 3.5

Ex: Given that cos θ = 3/5 find all the possible coordinatesfor point P(x,y) in the unit circle

Determine which quadrant θ will be in

The ratiois positive

Find the coordinates using sine & cosine

The possible coordinates for P are:

Find the angles

Practice: Given that show the angle in standardposition, and the coordinates of P on the unit circle.

Determine which quadrant θ will be in

The ratiois negative

Find the coordinates using sine & cosine

The possible coordinates for P are:

Find the angles

Ex: Given that sin θ = and find the exactvalue of the coordinates of point P(x,y) in the unit circle

Determine which quadrant θ will be in

The ratiois negative

Use the ratio to draw a right triangle

Find Base (Pythagorus)

Find the coordinates using sine & cosine

Coordinates of P:

Ex: Given that tan θ = 3/5 find all the possible coordinatesfor point P(x,y) in the unit circle

Determine which quadrant θ will be in

The ratiois positive

Find the coordinates using sine & cosine

The possible coordinates for P are:

Find the angles

HOMEWORK:

Assignment 3.2