1.6a day 1 Evaluating Trig Functions

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2008

Church of St. Mary, by Sir Christopher Wren, London, England 1672now at the National Churchill Museum, Westminster CollegeFulton, Missouri

When you were first introduced to trigonometry, trig functions were defined in terms of the sides of a right triangle.

sin

oppositehypotenuse

adjacentcos

tan If we superimpose this triangle on a unit circle, we can redefine the functions in terms of x and y.

opposite

hypotenuse

adjacent

hypotenuse

opposite

adjacent

Using these definitions, we can evaluate trig functions for angles on the x and y axis that do not make a triangle.

opposite

adjacent

sin

cos

x

y

0

2

3

2

x

y1

sin

cos

tan

opposite

hypotenuse

adjacent

hypotenuse

1

yy

1

xx

y

x

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

x

y

00

2

2

3

2

3

2

2

2

A sine curve is just the graph of the y values of a unit circle as the angle changes:

This means that we can use a unit circle to evaluate the sine function. All we have to do is find the y values!

(A unit circle has a radius of 1.)

x

y

0

2

3

2

2

The sine of an angle is the y-value (vertical distance).

If you forget, sine is vertical, like a stop sign!

sin 0 0

sin2

1

sin 0

3sin

2

1

sin 2 0

x

y

0

2

3

2

2

The cosine of an angle is the x-value (horizontal distance).

cos 0 1

cos2

0

cos 1

3cos

2

0

cos 2 1

x

y

0

2

3

2

2

Once you know sine and cosine, you can find the other functions using the definitions:

sintan

cos

coscot

sin

1sec

cos

1csc

sin

Functions are undefined when the denominator is zero.

x

y

0

2

3

2

2

Now let’s try a few.

Answer as quickly as you can!

tan 0 0

tan2

undefined

tan 0

3tan

2

undefined

tan 2 0

x

y

0

2

3

2

2

Now let’s try a few.

Answer as quickly as you can!

cot2

0

sec 0 1

sec2

undefined

3cot

2

0

3sec

2

undefined

csc 0 undefined

x

y

0

2

3

2

2

Now let’s try a few.

Answer as quickly as you can!

csc2

1

sec 1

csc undefined

cot 0 undefined

cot undefined

sec2

undefined