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Trigonometric Functions

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.

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Trigonometric Functions

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  1. Trigonometric Functions Unit Circle Approach

  2. The Unit Circle • Definition

  3. 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

  4. Six Trigonometric Functions of t

  5. Six Trigonometric Functions of t

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

  7. 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

  8. 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)

  9. Finding the Exact Values of the 6 Trig Functions of Quadrant Angles • Table with all of values on p. 387

  10. 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

  11. 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.

  12. 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)

  13. 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.

  14. 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

  15. 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

  16. 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.

  17. 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.

  18. Six Trig Functions • Tutorials • More Tutorials

  19. 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. 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

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