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

Finding the Values on Unit Circle Find the values of the six trig functions given the point on the unit circle

Six Trigonometric Functions of the Angle θ If θ = t radians, the functions are defined as: sin θ = sin tcsc θ = csc t cos θ = cos tsec θ = sec t tan θ = tan tcot 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 v 0 is a parabola. The range of the projectile, that is the horizontal distance that the projectile travels, is found by using the formula

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