Trigonometry Practice Problems

Here you will find a concise collection of trigonometry practice problems. Visit this page directly by visiting hunkim.com/trigonometry

Trigonometry

• Trigonometry Basic facts – hunkim.com/trigbasics
  
  
  
• Trigonometry – SOH CAH TOA balloon question
  Find length AB in the triangle below:
  
  
• Trigonometry – sin(A) vs angle A and meaning of reciprocal trig functions
  See right triangle below:
  
  1. Find \sin \theta
  2. Find \cos \theta
  3. Find \tan \theta
  4. Find \theta with a calculator
  5. Find an expression for \theta without a calculator
  6. Find an expression for \cos^{-1} \left(\frac{5}{13}\right) in terms of \theta
     
• Trig of an unknown special angle
  Solve \sin A=-\frac{\sqrt{2}}{2}, 0\leq A\leq 360^\circ
  
• Sine Law
  \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}. Prove that \frac{\sin A}{a}=\frac{\sin B}{b}
  1. Given the previous proof, why does it follow that:
  2. \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}?
  3. When do we use the Cosine Law instead of the Sine Law?
     
• Trigonometry – Sketch the hidden unit circle equation
  Sketch: y^2=1-x^2
  
• Triangles within circles
  Given the radius of a circle is 3, what is the value of the y-coordinate of a point on a circle in terms of \theta?
  
• Geometric meaning of sin cos and tan functions:
  See the unit circle below. What is the geometric meaning of:
  
  1. \sin \theta
  2. \cos \theta
  3. \tan \theta
     
• Sketch the angle in standard position (quadrant, reference angle, co-terminal angle)
  \theta=-300^\circ
  1. Sketch in standard position.
  2. Which quadrant is \theta in?
  3. Reference angle?
  4. First positive co-terminal angle?

Overview

• Basics and SOH CAH TOA
• Trigonometry in Standard Position
• Sine and Cosine Law
• Special Angles
• Basic Trigonometric Equations
• Unit Circle
• Trigonometric Identities

Basics and SOH CAH TOA

1. Solve x and y in the diagram below:
   
   
2. Is \theta in the triangle below 90^\circ?
   
3. Find \theta in the diagram below:
   
4. Find x in the triangle below:
   
   
5. See triangle below:
   
   1. Find x
   2. Find \angle a
   3. Find \angle b
   4. Permimeter?
   5. Area?
      
6. See the special triangle below:
   
   1. Find x
   2. Find \theta
      
7. See triangle below:
   
   1. Use the correct mathematical symbol to indicate that \angle B=90^\circ
   2. By convention, angles in triangles are in uppercase. By convention, how should you label the sides of the triangle?
   3. Is \angle BCA a right angle, acute, or obtuse?
      
8. How tall is the tree below?
   
   
9. Find \theta in the diagram below:
   
   
10. Find the height of the hill below:
    
    
11. The height of the pyramid below is 6.
    
    1. Find \theta
    2. Find \alpha

Trigonometry in Standard Position

1. Sketch \theta=30^\circ in standard position.
2. Locate Quadrants I to IV.
3. \theta=700^\circ
   1. Sketch in standard position.
   2. Reference angle?
4. Given \pi radians equals 180^\circ
   1. Sketch \theta=\frac{\pi}{6} in standard position.
   2. Sketch \theta=\frac{\pi}{4} in standard position.
   3. Evaluate \sin 30^\circ on your calculator in Degree mode
   4. Evaluate \sin \frac{\pi}{6} in Radian mode

Sine and Cosine Law

1. Solve the ASA triangle below:
2. 
3. Solve the following AAS triangle below:
   
4. Solve x and y in the SSA triangle below:
   
5. Ambiguous Case: \angle C=33^\circ. Side c=6 and side b=10.
   1. What are the possible angles of B?
   2. What are the possible lengths of a?
6. How many possible triangles?
   1. Given \sin B=1.2
   2. \angle A=30^\circ, a=10, and b=16
   3. \angle A=30^\circ, a=20, and c=16
   4. \angle A=30^\circ, a=7, and b=16
7. Enrichment: Prove the Cosine Law c^2=a^2+b^2-2ab\cos C
8. When is the Cosine Law used instead of the Sine Law?
9. Find x and y in the following SAS triangle below:
   
10. Solve a and b in the following SSS triangle below:
    
11. Does the Sine and Cosine Law work on right-angled triangles?
12. Starting at home, you jog N30^\circ W for 10 km. You then run S40^\circ W for 3 km. You then run towards home. How far did you jog?
13. Below is the graph of f(\theta)=\sin \theta or f(x)=\sin x
    
14. Evaluate f(30^\circ)
    1. Domain?
    2. Range?
    3. According to this graph \sin 30^\circ is equivalent to what?
15. Sketch y=\tan x

Special Angles

1. Evaluate without a calculator:
   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
2. Evaluate the following special angles:
   1. \sin 120^\circ
   2. \cos 135^\circ
   3. \tan (-690^\circ)
   4. -\sin 225^\circ
3. Evaluate the following quadrantal angles:
   1. \sin 180^\circ
   2. \cos(-180^\circ)
   3. \sin 270^\circ
   4. \tan(360^\circ)
4. If \sin \theta is negative and \cos \theta is positive, what quadrant must \theta be in?

Basic Trigonometric Equations

1. Solve the following trigonometric equations with the domain 0\leq\theta\leq360^\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=\pi
   9. \cos A=-\frac{12}{13}. Given 180^\circ<A<270^\circ find \tan A

Unit Circle

1. Sketch
   1. The unit circle: x^2+y^2=1
   2. x^2+y^2=9
   3. x^2+y^2=5
   4. (x-2)^2+(y-2)^2=4

1. Express the x and y coordinates  of a point on the unit circle in terms of the basic trigonometric ratios.
2. Evaluate (\sin^2 x+\cos^2 x)^2
3. Label the (x,y) coordinates on the unit circle: P(\theta):
   1. \theta=30^\circ
   2. \theta=120^\circ
   3. \theta=-135^\circ
   4. \theta=90^\circ
   5. \theta=-2700^\circ
4. \theta in standard position on the unit circle has coordinates \left(-\sqrt{3},1\right). Find \theta.

Trigonometric Identities

1. \tan\theta=\frac{\sin\theta}{\cos\theta}
2. \sin^2\theta+\cos^2\theta=1

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