Mad Slide Ruling
  
TRIGONOMETRY  
line decor
  HOME  ::  LESSONS  ::  INTERMEDIATE  ::
line decor
     
 
TRIGONOMETRICAL RATIOS FOR LARGE ANGLES ...

Objective:

To find trigonometrical ratios of large angles.

General:

  • In some problems obtuse or reflex angles may be used.  The slide rule does not show any angle past 90°.
  • To find trigonometrical ratios of larger angles, we us the following diagram and convert them to their equivalent acute angle in value; often known as the CAST rule.
Trigonometric Ratios for Large Angles (Lesson) 
  • The letters stand for the ratio which is positive in that quadrant.  So A means all ratios are positive in the first quadrant (acute angles); S means that only sine is positive in the second quadrant (obtuse angles) all other ratios are negative; T means that only tangent is positive in the third quadrant (reflex angles) all other ratios are negative and C means that only cosine is positive in the fourth quadrant (reflex angles) all other ratios are negative.
  • Cosecant obeys the same rules as sine, cotangent those of tangent and secant those of cosine.

 

Method:

  • Determine which quadrant the angle lies in.
  • Determine the acute angle made with the x-axis.
  • Determine the sign according to the CAST rule.
  • Find the trigonometrical ratio of this new angle (use the same trigonometrical ratio).

 

Example 1:  Calculate sin 225°.

  • The angle lies in the third quadrant.
  • The acute angle with the x-axis is 45°.
  • Sine is negative in the third quadrant.
  • Hence sin 225° is equivalent in value to -sin 45°.
  • Answer: -0.707

 

Example 2:  Calculate cos 335.6°.

  • The angle lies in the fourth quadrant.
  • The acute angle with the x-axis is 24.4°.
  • Cosine is positive in the fourth quadrant.
  • Hence cos 335.6° is equivalent in value to cos 24.4°.
  • Answer: 0.91

 

Example 3:  Calculate tan 140°.

  • The angle lies in the second quadrant.
  • The acute angle with the x-axis is 40°.
  • Tangent is negative in the second quadrant.
  • Hence tan 140° is equivalent in value to -tan 40°.
  • Answer: -0.839

 

Practice Questions:  Calculate the following trigonometrical ratios.

          1. sin 203° (same value as -sin 23°; Ans: -0.391)
          2. sin 310.5° (same value as -sin 49.5°; Ans: -0.76)
          3. cos 145.6° (same value as -cos 34.4°; Ans: -0.825)
          4. cos 226° (same value as -cos 46°; Ans: -0.695)
          5. tan 210° (same value as tan 30°; Ans: 0.577)
          6. tan 341.2° (same value as -tan 18.8°; Ans: -0.34)