Lesson 7-3Lesson 7-3

The Sine and Cosine Functions

Objective:

Objective:

To use the definitions of sine and cosine to find values of these functions and to solve simple

trigonometric equations.

Trigonometry

Trigonometry

• The sine function is abbreviated sin

Trigonometry

• The sine function is abbreviated sin

• The cosine function is abbreviated cos

Let P(x,y) be any point on the circle x2 + y2 = r2 and θ be

an angle in standard position with terminal ray OP, as

shown below.P(x,y)

r θ

O

Let P(x,y) be any point on the circle x2 + y2 = r2 and θ be

an angle in standard position with terminal ray OP, as

shown below.

• We define the sin θ by:

P(x,y)

r θ

O

Let P(x,y) be any point on the circle x2 + y2 = r2 and θ be

an angle in standard position with terminal ray OP, as

shown below.

• We define the sin θ by:

P(x,y)

r θ

O

Let P(x,y) be any point on the circle x2 + y2 = r2 and θ be

an angle in standard position with terminal ray OP, as

shown below.

• We define the sin θ by:

• We define the cos θ by:

P(x,y)

r θ

O

Let P(x,y) be any point on the circle x2 + y2 = r2 and θ be

an angle in standard position with terminal ray OP, as

shown below.

• We define the sin θ by:

• We define the cos θ by:

P(x,y)

r θ

O

If the terminal ray of an angle θ in standard position

passes through (-3,2), find sin θ and cos θ.

Unit Circle

Unit Circle

• The unit circle is a circle with a center at the origin and has a radius of 1.

Unit Circle

• The unit circle is a circle with a center at the origin and has a radius of 1.

• Therefore its equation is simply:

Unit Circle

• The unit circle is a circle with a center at the origin and has a radius of 1.

• Therefore its equation is simply:

Unit Circle

Unit Circle

• Which now allows us to take our two formulas for sin θ and cos θ and change them to:

Unit Circle

• Which now allows us to take our two formulas for sin θ and cos θ and change them to:

Unit Circle

• Which now allows us to take our two formulas for sin θ and cos θ and change them to:

Now angles of rotations can locate you anywhere in the four quadrants. Since sin θ can now be determined strictly by the y-values, that means

the sine of an angle will always be positive if your angle

of rotation locates you in the 1st of 2nd quadrants.

Now angles of rotations can locate you anywhere in the four quadrants. Since sin θ can now be determined strictly by the y-values, that means

the sine of an angle will always be positive if your angle

of rotation locates you in the 1st of 2nd quadrants.

+ +

- -

Likewise, cos θ can now be determined by the x-values,

so the cosine function will always be positive if the angle of rotation locates

you in the 1st or 4th quadrants.

Likewise, cos θ can now be determined by the x-values,

so the cosine function will always be positive if the angle of rotation locates

you in the 1st or 4th quadrants.

+

+-

-

Find:

Find:

Find:

Find:

From the previous examples and the definitions of sin θ and cos θ we can

see that the sine and cosine functions repeat their values every 360° or 2π radians. Formally, this

means that for all θ:

From the previous examples and the definitions of sin θ and cos θ we can

see that the sine and cosine functions repeat their values every 360° or 2π radians. Formally, this

means that for all θ:

From the previous examples and the definitions of sin θ and cos θ we can

see that the sine and cosine functions repeat their values every 360° or 2π radians. Formally, this

means that for all θ:

We summarize these facts by saying that the sine and cosine functions are

periodic and that they both have a fundamental period of

3600 or 2π radians.

Assignment:Assignment:

Pgs. 272-274Pgs. 272-2741-41 odd, omit 311-41 odd, omit 31