2.4 Trigonometry 1

• Sides of a right-angled triangle: hypotenuse, opposite and adjacent
• Sine, cosine and tangent ratios
• Finding unknown sides using trigonometric ratios
• Finding unknown angles using inverse ratios
• Angles of elevation and depression

• identify the hypotenuse, opposite and adjacent sides relative to an angle
• define sine, cosine and tangent of an acute angle
• use trigonometric ratios to find unknown sides
• use inverse ratios to find unknown angles
• apply trigonometry to angles of elevation and depression

“How can one angle tell you a distance you cannot measure?”

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1. The three ratios
   40 min · 2 practice tasks · quiz
   Why a fixed angle gives a fixed ratio, and how to use that to find a missing side.
2. Finding angles, elevation and depression
   40 min · 2 practice tasks · quiz
   Working backwards from sides to angles, and applying it to real heights.
3. Labelling from either acute angle
   40 min · 1 practice tasks · quiz
   Opposite and adjacent swap when you work from the other acute angle; the hypotenuse does not.
4. Calculators and mathematical tables
   40 min · 1 practice tasks · quiz
   Get sine, cosine and tangent of an acute angle from a scientific calculator in degree mode, and from four-figure tables.
5. When the unknown is the denominator
   40 min · 1 practice tasks · quiz
   Rearrange a trigonometric ratio so that the missing side, sitting under the fraction line, is found by dividing.
6. Sine and cosine of complementary angles
   40 min · 1 practice tasks · quiz
   sin theta equals cos(90 minus theta), because opposite of one acute angle is adjacent of the other.
7. Relating sine, cosine and tangent
   40 min · 1 practice tasks · quiz
   tan theta is sin theta over cos theta, and sin squared plus cos squared is 1, both from the same right-angled triangle.
8. Special angle 45 degrees
   40 min · 1 practice tasks · quiz
   An isosceles right-angled triangle gives the exact ratios for 45 degrees without a calculator.
9. Special angles 30 and 60 degrees
   40 min · 1 practice tasks · quiz
   An equilateral triangle of side 2, split by an altitude, gives the exact ratios for 30 and 60 degrees.
10. Special angle 90 degrees and the exact-value table
    40 min · 1 practice tasks · quiz
    sin 90 is 1, cos 90 is 0, tan 90 is undefined, and the five-angle table is the one to know by heart.
11. Inverse ratios in practice
    40 min · 1 practice tasks · quiz
    Find an acute angle from two sides using inverse sine, cosine or tangent, then check that the two acute angles add to 90 degrees.
12. Solving a complete right-angled triangle
    40 min · 1 practice tasks · quiz
    From one side and one acute angle, or from two sides, find every other side and angle.
13. Height from two positions
    40 min · 1 practice tasks · quiz
    Two angles of elevation from points a known distance apart give the height of an object you cannot reach.
14. Depression from a height
    40 min · 1 practice tasks · quiz
    An angle of depression from a cliff, tower or window equals the angle of elevation from below; use it to find distances along the ground.
15. A clinometer and trigonometry around us
    40 min · 1 practice tasks · quiz
    Make a simple clinometer, measure a real height, and see trigonometry in roofs, ramps, surveying and Kenyan careers.