0580

Right Angled Triangles

Pythagoras and Trigonometry

The values to know

  • For a handful of angles the ratios can be written exactly, as a fraction or a surd rather than a rounded decimal
A right-angled isosceles triangle with both short sides 1 and, by Pythagoras, a hypotenuse of root 2, giving the exact values sin 45 = 1 over root 2, cos 45 = 1 over root 2 and tan 45 = 1
Source: Exact trig values by Save My Exams
An equilateral triangle of side 2 cut in half to give a right-angled triangle with sides 1, 2 and root 3, from which the exact values follow: sin 30 is a half, cos 30 is root 3 over 2, tan 30 is 1 over root 3, sin 60 is root 3 over 2, cos 60 is a half and tan 60 is root 3
Source: Exact trig values by Save My Exams
θ0°30°45°60°90°
sin θ0½√2 ÷ 2√3 ÷ 21
cos θ1√3 ÷ 2√2 ÷ 2½0
tan θ0√3 ÷ 31√3—
  • The syllabus requires sine and cosine at 0°, 30°, 45°, 60° and 90°, and tangent at 0°, 30°, 45° and 60° only
  • Tangent has no value at 90°, which is why the syllabus stops short of it there
  • The sine row read forwards matches the cosine row read backwards, which halves what there is to remember
  • √2 ÷ 2 is the same as 1 ÷ √2, and √3 ÷ 3 is the same as 1 ÷ √3, so either form is correct
  • These come up whenever a question asks for an exact value, which rules out a rounded decimal
Exam tip

Neither "Pythagoras" nor "hypotenuse" appears in any of the 43 papers — what appears is a marked right angle and two known values. Write the ratio you chose with the values substituted before rearranging: that line is the method mark. Angles are given to 1 decimal place and lengths to 3 significant figures unless the question says otherwise, and add for a hypotenuse but subtract for a shorter side.

Worked example

Using the exact values

Without a calculator, find the exact value of sin 60° × cos 30°, and of tan 45° + sin 30°.

Solution:

  • These come from the two triangles worth memorising, so no calculator is needed
  • sin 60° = √3/2 and cos 30° = √3/2
  • Multiplying: (√3/2) × (√3/2) = (√3 × √3) / (2 × 2)
  • √3 × √3 = 3, so the answer is 3/4
  • tan 45° = 1 and sin 30° = 1/2
  • Adding: 1 + 1/2 = 3/2, or 1.5
  • Both answers are exact, so leaving them as fractions is the answer — 0.75 would be a decimal approximation of an exact value