Grade 10

Trigonometry in the Right Triangle

Sine, cosine and tangent built out of similar triangles, then used to reach sides, angles and heights that cannot be measured directly.

6 lessons

  1. 1Why does the ratio between sides depend only on the angle?Two right triangles sharing an acute angle are similar, and in similar triangles the ratio between any two corresponding sides is equal. So the ratio does not depend on the size of the triangle but only on the angle — which is why it can be written in a single table and used for every triangle.30m
  2. 2What are sine, cosine and tangent?All three are ratios between sides of a right triangle. Sine is the opposite side over the hypotenuse, cosine is the adjacent side over the hypotenuse, and tangent is the opposite over the adjacent. Sine and cosine are always between zero and one, because the hypotenuse is the longest side.35m
  3. 3How do you find a side length when the angle is known?Pick the ratio linking the side you want to the side you have: sine if a hypotenuse and an opposite side are involved, cosine for a hypotenuse and an adjacent side, tangent for two legs. Write the ratio, substitute what is known, and isolate the unknown. The angle stays inside the function until the very end.35m
  4. 4How do you find an angle when two sides are known?Work out the ratio between the two known sides, then apply the inverse function on the calculator — inverse sine, inverse cosine or inverse tangent, according to which sides they are. The ratio is a number, and the inverse function translates it back into the angle that produced it.35m
  5. 5What are the special angles and what is the fundamental trigonometric identity?The special angles are thirty, forty-five and sixty degrees, whose ratios come from two simple triangles and are worth knowing by heart. The fundamental identity says that sine squared plus cosine squared equals one — it follows directly from Pythagoras' theorem and lets you find one ratio from the other.40m
  6. 6What is the difference between an angle of elevation and an angle of depression?Both are measured from the horizontal. An angle of elevation is the angle you raise your line of sight through to look at something higher; an angle of depression is the angle you lower it through to look at something lower. Between two observers looking at each other the two angles are equal, because they are alternate angles between parallel horizontals.40m