BC Pre-Calculus 11 Trigonometry

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Last year you learned about right-angle trigonometry: SOH CAH TOA. This year in this Pre-Calculus 11 you will learn how to solve non right-angle triangles using the Sine Law and the Cosine Law. Do your best to understand this year’s trigonometry concepts because you will learn more about trigonometry next year. 

  • Use of sine and cosine laws to solve non-right triangles, including ambiguous cases
  • Contextual and non-contextual problems
  • Angles in standard position
  • Degrees
  • Special angles, as connected with the 30-60-90 and 45-45-90 triangles
  • Unit circle
  • Reference and co-terminal angles
  • Terminal arm
  • Trigonometric ratios
  • Simple trigonometric equations
  1. Label the location of the four quadrants
  2. In which Quadrant is \theta located?
    1. \theta=120^\circ
    2. \theta=-45^\circ
    3. \theta=400^\circ
    4. \theta=-1100^\circ
  3. \theta=300^\circ
    1. What is the reference angle?
    2. Find a positive coterminal angle to \theta=300^\circ
    3. Find a negative coterminal angle to \theta=300^\circ
  4. Enrichment: Radians vs. Degrees
    This year we measure the angle \theta in degrees. Next year we use a different unit called radians. One full revolution is 2\pi which is equivalent to 360^\circ.
    1. Convert \pi radians to degrees
    2. Convert \frac{\pi}{4} radians to degrees
    3. Convert \frac{\pi}{6} radians to degrees
  5. Enrichment:
    1. Show that the area of a triangle is A_{\triangle}=\frac{1}{2}ab \sin C
    2. Find the area of the triangle below:

  6. Enrichment
    1. Use the triangle below to help you prove the Sine Law:
      \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}
    2. Giving the previous proof, why does it follow that
      \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}?
  7. Solve x and y the ASA triangle below:
  8. Solve x and \theta the following AAS triangle below:

  9. Solve \theta and a the following SSA triangle below:

  10. No diagram: Solve the following triangle:
    \angle C=140^\circ, b=6, c=30
  11. Consider the ambiguous case:
    \angle C=33^\circ. Side c=6. Side b=10.
    1. What are the possible angles of B?

    2. What are the possible lengths of a?
  12. Enrichment: State the number of possible triangles that can be formed. Confirm your answer with an online triangle calculator.
    1. \angle B=30^\circ, a=27, b=22
    2. \angle B=96^\circ, b=25, a=6
    3. \angle B=34^\circ, a=23, b=7
    4. \angle A=30^\circ, AC=8, BC=5
  13. Find side length CB in the diagram below:

  14. Find the largest possible angle in the diagram below:

  15. Given c^2=a^2+b^2-2ab \cos C, find expression for \angle C.
  16. Unit circle:
    1. Equation of the unit circle?
    2. Enrichment: What is the equation of a circle with radius r centered at the origin?
    3. Sketch the unit circle
    4. Explain why y=\sin \theta on the unit circle
    5. Explain why x=\cos \theta on the unit circle
    6. Enrichment: Where does the trigonometric identity \sin^2\theta + \cos^2\theta=1 come from?
  17. Enrichment: Use your knowledge of the primary trigonometric ratios and the Pythagorean Theorem to prove the Cosine Law:
    c^2=a^2+b^2-2ab \cos C

  18. When solving a non-right-angled triangle, when should the Sine Law vs. Cosine Law be used?
  19. Find x and y in the following SAS triangle:

  20. Find a and b in the following SSS triangle:

  21. See the right triangle below:
    1. Solve x using the Pythagorean Theorem
    2. Find the value of the missing angle
    3. Solve x using the Sine Law
    4. Solve x using the Cosine Law
    5. Solve x using the Cosine Law
  22. Solve the unknown angles in the diagram below:
    1. Using SOH CAH TOA
    2. Using similar triangles and your knowledge of a special triangle
    3. Using the Sine Law
    4. Using the Cosine Law
  23. Solve x in the triangle below:
  24. Basic trigonometric identity: \tan \theta=\frac{\sin \theta}{?}
  25. Explain how drawing a 2-2-2 equilateral can help you memorize the primary trigonometric ratios.
  26. Memorize the values of the following special angles:
    1. \sin 30^\circ
    2. \sin 45^\circ
    3. \sin 60^\circ
    4. \cos 30^\circ
    5. \cos 45^\circ
    6. \cos 60^\circ
    7. \tan 30^\circ
    8. \tan 45^\circ
    9. \tan 60^\circ
  27. Find the exact height of the tree without a calculator and
    simplify your answer using your knowledge of special angles.

  28. Evaluate
    1. \sin 120^\circ
    2. \cos 330^\circ
    3. \sin 225^\circ
    4. -\sin 225^\circ
    5. \tan(-420^\circ)
  29. Quadrantal angles – Find:
    1. \sin 90^\circ
    2. \cos 180^\circ
    3. \sin(-360)^\circ
    4. \tan(180)^\circ
  30. Label the (x,y) coordinates on the unit circle for P(\theta) when:
    1. \theta=30^\circ
    2. \theta=45^\circ
    3. \theta=60^\circ
    4. \theta=90^\circ
    5. \theta=210^\circ
    6. \theta=270^\circ
    7. \theta=315^\circ
    8. \theta=720^\circ
  31. If \sin \theta is negative and \cos \theta is positive, what quadrant must \theta be in?
  32. Solve the following trigonometric equations within the domain 0\leq \theta \leq 360^\circ:
    1. \sin \theta=\frac{1}{2}
    2. \sin \theta=-\frac{1}{\sqrt{2}}
    3. \sin A=\frac{\sqrt{2}}{2}
    4. \sin \beta=\frac{-\sqrt{3}}{2}
    5. \cos \theta=-0.5
    6. \tan x=\sqrt{3}
    7. \tan \theta=-2
    8. \sin \theta=2
  33. \theta in standard position on the unit circle has coordinates \left(-\frac{\sqrt{3}}{2},\frac{1}{2}\right). Find \theta
  34. Challenge: A boat travels 13 km in the direction N30^\circ W. It then adjusts its course and heads S70^\circ W, travelling antoher 20 km in this new direction.
    1. How far is the boat from its initial position?
    2. Enrichment:  Bearings are angles measured in a clockwise direction from the north line.  What is the bearing of the boat in its final position as compared to its initial position?
  35. Enrichment: Visually represent \tan \theta on the unit circle.