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Comprehensive Review on Trigonometry for Any Angle - Day 41 Plan and Homework

In this session, we will review key concepts from right triangle trigonometry, the unit circle, and trigonometric functions for any angle. We'll discuss the properties of even and odd functions, including examples of sine, cosine, and tangent. Students will also engage in exercises related to reference angles, quadrant determination, and evaluating trigonometric functions based on points in different quadrants. Homework from Section 4.4 will be assigned to reinforce today's topics.

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Comprehensive Review on Trigonometry for Any Angle - Day 41 Plan and Homework

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  1. Pre Calculus Trigonometry for Any Angle Day 41

  2. Plan • Quiz – Right Triangle Trig • Review of Homework • Section 4.2 • Review the Unit Circle • 4.4 Trig For Any Angle • Homework 4.4 P 297-298 # 3, 7, 13, 17, 21, 23, 31, 35, 47, 53, 85, 88 • Quiz Unit Circle Next time

  3. Right Triangle Trigonometry

  4. Fill out some stuff we already know

  5. Fill out some stuff we already know

  6. Fill out some stuff we already know

  7. Fill out some stuff we already know

  8. Even and Odd Trig Functions An even function: f(x) = f(-x) cos(30o) = cos(-30o)? cos(135o) = cos(-135o)? The cosine and its reciprocal are evenfunctions.

  9. Even and Odd Trig Functions An odd function: f(-x) = -f(x) sin(-30o) = -sin(30o)? sin(-135o) = -sin(135o)? The sine and its reciprocal are oddfunctions.

  10. Even and Odd Trig Functions An odd function: f(-x) = -f(x) tan(-30o) = -tan(30o)? tan(-135o) = -tan(135o)? The tangent and its reciprocal are oddfunctions.

  11. Even and Odd Trig Functions Cosine and secant functions are even cos (-t) = cos t sec (-t) = sec t Sine, cosecant, tangent and cotangent are odd sin (-t) = - sin t csc (-t) = - csc t tan (-t) = - tan t cot (-t) = - cot t Add these to your worksheet

  12. Unit Circle Review How can we memorize it? • Symmetry • For Radiansthedenominatorshelp! • Knowing the quadrant givesthe correct+ / - sign

  13. Practice… Get out your Unit Circle, Pencil and Paper! • ON YOUR OWN try these… • Write the question and the answer

  14. Check your work

  15. How did you do??

  16. Reference Angles Let θ be an angle in standard position. Its reference angle is the acute angle θ’ (called “theta prime”) formed by the terminal side of θ and the horizontal axis.

  17. Look at these

  18. Reference Angles Let θ be an angle in standard position and its reference angle has the same absolute value for the functions, the sign ( +/ - ) must be determined by the quadrant of the angle. • Quadrant II θ’ = π – θ (radians) = 180o – θ (degrees) • Quadrant III θ’ = θ – π (radians) = θ – 180o (degrees) • Quadrant IV θ’ = 2π – θ (radians) = 360o – θ (degrees)

  19. r θ (x,y) Trigonometry for any angle Given a point on the terminal side Let  be an angle in standard position with (x, y) a point on the terminal side of  and r be the length of the segment from the origin to the point Then….

  20. Trig for any angle • The six trigonometric functions can be defined as Add these definitions to summary worksheet

  21. Your Chart should look like…

  22. (-3, 4) r θ Evaluating Trig Functions: Find sin, cos and tan given (-3, 4) is a point on the terminal side of an angle. • Find r • Find the ratio of the sides of the reference angle • Make sure you have the correct sign based upon quadrant Find r. (-3)2 + (4)2= r2 r =5 sin θ = 4/5 cos θ = -3/5 tan θ = -4/3

  23. Name the quadrant… A little different twist… The cosine and sine of the angle are positive 1 The cosine and sine are negative 3 The cosine is positive and the sine is negative. 4 The sine is positive and the tangent is negative 2 The tangent is positive and the cosine is negative. 3 The secant is positive and the sine is negative. 4

  24. θ r (4, -5) Look at this one… Given tan  = -5/4 and the cos  > 0, find the sin  and sec . Which quadrant is it in? The tangent is negative, and the cosine is positive Quadrant IV at point (4, -5) Find r and use the triangle to find the sine and secant

  25. (x, 1) 3 θ And this one… • Let  be an angle in quadrant II such that sin  = 1/3 find the cos  and the tan . Set up a triangle based upon the information given . Calculate the other side Find the other trigonometric functions

  26. Homework 22 Section 4.4 pp. 297-298; 3, 7, 13, 17, 21, 23, 31, 35, 47, 53, 85, 88

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