Trigonometric Functions

Unit Circle Approach

The Unit Circle

Definition

Six Trigonometric Functions of t

The sine function associates with t the y-coordinate of P and is denoted by

sin t = y

The cosine function associates with t the x-coordinate of P and is denoted by

cos t = x

Six Trigonometric Functions of t

0, the tangent function is defined as

tan

If x

ytx

0, the cosecant function is defined as

1csc

If y

ty

Six Trigonometric Functions of t

0, the secant function is defined as

1sec

If x

tx

0, the cotangent function is defined as

sec

If y

xty

Finding the Values on Unit Circle

Find the values of the six trig functions given the point on the unit circle

5 2,

3 3

Six Trigonometric Functions of the Angle θ

If θ = t radians, the functions are defined as: sin θ = sin t csc θ = csc t cos θ = cos t sec θ = sec t tan θ = tan t cot θ = cot t

Finding the Exact Values of the 6 Trig Functions of Quadrant Angles

Unit Circle – radius = 1 Quadrant Angles: 0, 90, 180, 270, 360 degrees 0, π/2, π, 3π/2, 2π Point names at each angle: (1, 0) (0, 1) (-1, 0) (0, -1)

Finding the Exact Values of the 6 Trig Functions of Quadrant Angles

Table with all of values on p. 387

Circular Functions

A circle has no beginning or ending. Angles on a circle therefore have many names because you can continue to go around the circle.

Positive Angles Negative Angles

Finding Exact Values

Reminder of how to use your hand to find the value of a trig function for 0, 30, 45, 60, or 90 degree reference angles

Reminder of how to use your hand to find the value of a trig function for 0, pi sixths, pi fourths, pi thirds and pi halves reference angles.

Finding Exact Values

Angles in Radians: 1. Determine reference angle 2. Change fraction to mixed numeral 3. Determine quadrant 4. Determine value using hand 5. Determine whether value is positive or

negative in that quadrant (All Scientists Take Calculus)

Finding Exact Values - Degrees

If Angle is in degrees we will need to determine our reference angle first by using the following rules:

If the angle is in the first quadrant – it is a reference angle

If the angle is in the second quadrant – subtract the angle from 180.

Finding Exact Values - Degrees

If the angle is in the third quadrant – subtract 180 from the angle

If the angle is in the fourth quadrant – subtract the angle from 360

Finding Exact Values - Degrees

(1) Determine whether value is positive or negative from the quadrant

(2) Find reference angle – using preceding rules

(3) Determine value of function using hand

Using Calculator to Approximate Value

If angle is not one that uses one of the given reference angles, calculator will be used to approximate the value.

This value is not exact as the previous values have been

Be careful that calculator is in correct mode.

Using a Circle of Radius R

To find the trig values given a point NOT ON THE UNIT CIRCLE

Be sure to read the directions before finding the six trig functions.

Six Trig Functions

Tutorials

More Tutorials

Applications – Projectile Motion

The path of a projectile fired at an inclination θ to the horizontal with initial speed v0 is a parabola. The range of the projectile, that is the horizontal distance that the projectile travels, is found by using the formula

20

2 2

sin(2 )

32.2 / s 9.9 /

vR

g

where g ft or m s

Applications – Projectile Motion

The projectile is fired at an angle of 45 degrees to the horizontal with an initial speed of 100 feet per second. Find the range of the projectile