Trigonometry

Sine, cosine and tangent

In a right triangle, sine, cosine and tangent link an angle with the ratio of two sides. Entrance tests use exact values and ratios rather than calculator decimals.

The rules

• sin α = opposite / hypotenuse; cos α = adjacent / hypotenuse; tan α = opposite / adjacent.
• Exact values: sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2, and cos is the same in reverse order.
• sin²α + cos²α = 1.
• If you know one ratio, find the third side with Pythagoras, then any other ratio.

Practice questions with solutions

Generated by the same engine as the Kollego course and solvable without a calculator. Pick an answer in your head, then open the solution.

1. 1. In a right triangle the hypotenuse is 40 and sin θ = 6/10. How long is the side opposite θ?

   • a32
   • b6
   • c16
   • d24
   Show solution

   Answer: 24

   sin θ = opposite / hypotenuse, so opposite = hypotenuse · sin θ.

   40 · (6/10) = 24.

2. 2. In a right triangle the hypotenuse is 50 and sin θ = 7/25. How long is the side opposite θ?

   • a36
   • b14
   • c7
   • d48
   Show solution

   Answer: 14

   sin θ = opposite / hypotenuse, so opposite = hypotenuse · sin θ.

   50 · (7/25) = 14.

3. 3. In a right triangle the hypotenuse is 26 and sin θ = 5/13. How long is the side opposite θ?

   • a16
   • b10
   • c24
   • d5
   Show solution

   Answer: 10

   sin θ = opposite / hypotenuse, so opposite = hypotenuse · sin θ.

   26 · (5/13) = 10.

4. 4. In a right triangle the hypotenuse is 20 and sin θ = 3/5. How long is the side opposite θ?

   • a8
   • b16
   • c3
   • d12
   Show solution

   Answer: 12

   sin θ = opposite / hypotenuse, so opposite = hypotenuse · sin θ.

   20 · (3/5) = 12.

5. 5. In a right triangle the hypotenuse is 39 and sin θ = 5/13. How long is the side opposite θ?

   • a5
   • b36
   • c15
   • d24
   Show solution

   Answer: 15

   sin θ = opposite / hypotenuse, so opposite = hypotenuse · sin θ.

   39 · (5/13) = 15.

6. 6. From point A on level ground the top of a tower is seen at an angle of elevation of 30°. Walking 38 m straight towards the tower to point B, the angle is 60°. How far is B from the foot of the tower (in m)?

   • a38
   • b19
   • c76
   • d38/3
   Show solution

   Answer: 19

   Two right triangles share the height h: tan 60° = h/x and tan 30° = h/(x + d).

   h = x·√3 and h = (x + 38)/√3 ⇒ 3x = x + 38 ⇒ x = 19.

Common mistakes

• Mixing up opposite and adjacent side.
• Using the trig ratios in a triangle without a right angle.
• Calculator in radians when the angle is in degrees.

Related topics

All maths topics · T-Kurs · M-Kurs · W-Kurs