Primary Trigonometric Ratios

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• Sine, cosine, and tangent ratios
• Right-triangle problems: determining missing sides and/or angles using trigonometric ratios and the Pythagorean theorem
• Contexts involving direct and indirect measurement

1. Explain the acronym SOH CAH TOA
   Solution
   \sin \theta=\frac{\text{opposite}}{\text{hypotenuse}}, \cos \theta=\frac{\text{adjacent}}{\text{hypotenuse}}, \tan \theta=\frac{\text{opposite}}{\text{adjacent}}
2. Use the correct mathematical notation to incidicate that \angle ABC=90^\circ
   Solution
   
   
3. By convention angles in triangles are in uppercase.  By convention, how should you label the sides of the triangle?
   Solution
   Lowercase letters are placed opposite the corresponding angle
4. What kind of triangles does SOH CAH TOA work with?
   Solution
   Right triangles
5. See triangle below:
   
   1. Find \sin \theta
      Solution
      \frac{3}{5}
   2. Find \cos \theta
      Solution
      \frac{4}{5}
   3. Find \tan \theta
      Solution
      \frac{3}{4}
   4. Find \sin \alpha
      Solution
      \frac{4}{5}
   5. Find \cos \alpha
      Solution
      \frac{3}{5}
   6. Express \cos \alpha as a decimal number
      Solution
      0.6
   7. Find \tan \alpha
      Solution
      \frac{4}{3}
   8. Find \theta in degrees
      Solution
      \approx 36.9^\circ
   9. Find \alpha in degrees
      Solution
      \approx 53.1^\circ
6. See triangle below:
   
   1. Find x
      Solution
      \sqrt{3}
   2. Find \theta
      Solution
      30^\circ
   3. Find \sin 60^\circ
      Solution
      \frac{\sqrt{3}}{2}
7. See the right triangle below:
   
   1. Find \sin \theta
      Solution
      \frac{5}{13}
   2. Find \cos \theta
      Solution
      \frac{12}{13}
   3. Find \tan \theta
      Solution
      \frac{5}{12}
   4. Find \sin \alpha
      Solution
      \frac{12}{13}
   5. Find \cos \alpha
      Solution
      \frac{5}{13}
   6. Find \tan \alpha
      Solution
      \frac{12}{5}
   7. Find \alpha using two different trigonometric ratios
      Solution
      \alpha=\cos^{-1}\left(\frac{5}{13}\right)=\tan^{-1}\left(\frac{12}{5}\right)\approx 67.4^\circ
8. How tall is the tree below?
   
   Solution
   10\tan 30^\circ=\frac{10\sqrt{3}}{3}\approx 7.77
9. Your friend is 6’ tall and stands 12 steps sway from the school.  Each step measures 2’.  Using a clinometer, you find that the roof of the school is at an angle of elevation of 45^\circ.  The diagram below is not to scale:
   
   1. Find the height of your school without using SOH CAH TOA.
      Solution
      30'
   2. Find the height of your school using SOH CAH TOA.
      Solution
      30'
10. How far apart is person A from person B?
    
    Solution
    33.2
11. Find length AB in the triangle below:
    
    Solution
    11.9

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