2.5 Trigonometry (II)

• Unit circle and trigonometric ratios from the unit circle
• Angles greater than 90 degrees and negative angles
• Radian measure; converting degrees to radians and back
• Graphs of sin x, cos x and tan x from −360 to 360 degrees
• Deriving the sine and cosine rules; solving triangles and applying the rules

• determine trigonometric ratios of angles using a unit circle
• determine trigonometric ratios of angles greater than 90 degrees in different situations
• determine trigonometric ratios of negative angles in different situations
• solve trigonometric equations involving trigonometric ratios in the range 0 to 360 degrees
• convert angles from degrees to radians and vice versa in different situations
• draw graphs of trigonometric functions in degrees and radians
• derive sine and cosine rules using triangles
• solve triangles using the sine and cosine rule
• appreciate use of trigonometry in real life situations

“Why is the knowledge of trigonometry important?”

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1. Ratios on the unit circle
   40 min · 1 practice tasks · quiz
   On a circle of radius 1, the coordinates of a point are the cosine and the sine of the angle.
2. Angles from 90 to 180 degrees
   40 min · 1 practice tasks · quiz
   In the second quadrant the related acute angle keeps the sine positive and makes the cosine and the tangent negative.
3. The third and fourth quadrants
   40 min · 1 practice tasks · quiz
   A full turn uses one sign pattern: all, sine, tangent, cosine, moving anticlockwise.
4. Negative angles and full turns
   40 min · 1 practice tasks · quiz
   A clockwise angle is negative, and adding or subtracting 360 degrees returns the same three ratios.
5. Equations from 0 to 360 degrees
   40 min · 1 practice tasks · quiz
   A single ratio usually has two solutions in a full turn, found from the related acute angle and the sign pattern.
6. Rearranging to one ratio
   40 min · 1 practice tasks · quiz
   An equation in a single ratio is rearranged or factorised first, and dividing by the ratio can drop solutions.
7. The radian
   40 min · 1 practice tasks · quiz
   One radian is the centre angle at which the arc equals the radius, and pi radians make a half turn.
8. Degrees and radians either way
   40 min · 1 practice tasks · quiz
   Multiply by pi/180 to reach radians, and by 180/pi to reach degrees.
9. Graphs of sine and cosine
   40 min · 1 practice tasks · quiz
   From -360 degrees to 360 degrees, sine and cosine are waves of height 1 that repeat every full turn.
10. The graph of tangent
    40 min · 1 practice tasks · quiz
    Tangent repeats every 180 degrees and breaks at the angles where cosine is 0.
11. The sine rule
    40 min · 1 practice tasks · quiz
    In any triangle, a side divided by the sine of the opposite angle is the same for all three pairs.
12. The cosine rule
    40 min · 1 practice tasks · quiz
    Two sides and the included angle give the third side, and three sides give an angle.