Glossary Checklist FAQ

E. Trigonometric Ratios for Any Angle


 

Investigation

Patterns in Trigonometric Ratios for Any Angle


Open the Reference Angle applet.

  1. Click on both “Show reference angle” and “Show side lengths” to make them appear.

  2. Investigate the points \((3, 4)\), \((-3, 4)\), \((-3, -4)\), and \((3, -4)\) by moving the red dot to those coordinates.

  3. Use a table similar to this to summarize your findings.

    4. Point Angle in Standard Position
    		Reference Angle Quadrant
    		Opposite
    		Adjacent
    		Hypotenuse
    \((3, 4)\)            
    \((-3, 4)\)            
    \((-3, -4)\)            
    \((3, -4)\)            

  4. What pattern(s) do you notice about the values?

  5. Try other points to see if the patterns hold.

  6. Determine the exact values of sin \(\theta\), cos \(\theta\), and tan \(\theta\) for each of the angles, using the information in the table from part 3. What do you notice?

    Point Angle in Standard Position, \(\theta\)
    		sin \(\theta\)
    		cos \(\theta\) tan \(\theta\)
    \((3, 4)\)        
    \((-3, 4)\)        
    \((-3, -4)\)        
    \((3, -4)\)        

 

 Key Lesson Marker


As you likely learned in the Investigation, a reference right triangle can be created in any quadrant using a point on the terminal arm of an angle in standard position. The reference triangle is always located adjacent to the \(x\)-axis. The primary trigonometric ratios for the resulting reference triangle can then be determined.

Primary Trigonometric Ratios, Given a Point on the Terminal Arm of an Angle in Standard Position

Given a point, \((x, y)\), on the terminal arm of angle \(\theta\) in standard position, at a distance \(r\) from the origin, a reference right triangle with a height of \(y\) units and a base length of \(x\) units can be constructed by drawing a vertical line from the point to the \(x\)-axis.

The primary trigonometric ratios in terms of \(x\), \(y\), and \(r\) can then be defined using the reference triangle.

\[\sin \theta = \frac{y}{r}, \thinspace \cos \theta = \frac{x}{r}, \thinspace \rm{and} \thinspace \tan \theta = \frac{y}{x}\]