Introduction to Trigonometry
Trigonometry uses three ratios, sine, cosine and tangent, to connect the angles and side lengths of a right-angled triangle: for an angle x other than the right angle, the hypotenuse is the longest side, the opposite side is across from x, and the adjacent side is next to x. The three ratios, sin(x) = opposite/hypotenuse, cos(x) = adjacent/hypotenuse and tan(x) = opposite/adjacent, often remembered using the mnemonic SOH CAH TOA, find a missing side or angle in a right-angled triangle.
Before you start
Make sure you're comfortable with these topics first:
Method
- Check the triangle has a right angle, then label the hypotenuse (the longest side, opposite the right angle).
- Relative to the angle you are using, label the opposite side (across from the angle) and the adjacent side (next to the angle, but not the hypotenuse).
- Decide which two sides are involved: hypotenuse and opposite means sine, hypotenuse and adjacent means cosine, opposite and adjacent means tangent (SOH CAH TOA).
- To find a missing side, write the ratio as an equation, substitute the known angle and side, then rearrange to make the missing side the subject.
- To find a missing angle, write the ratio using the two known sides, then use the inverse function on a calculator (sin^-1, cos^-1 or tan^-1) to find the angle.
- Round your final answer to an appropriate degree of accuracy (usually 1 decimal place), and check it is sensible: the side opposite the biggest angle should be the longest.
Worked example
In a right-angled triangle, the angle at A is 35 degrees, and the hypotenuse AC is 14 cm. Find the length of the side opposite angle A, to 1 decimal place.
- The opposite side and the hypotenuse are involved, so use sine: sin(A) = opposite / hypotenuse.
- Substitute the known values: sin(35) = opposite / 14.
- Rearrange to make the opposite side the subject: opposite = 14 x sin(35).
- Calculate: opposite = 14 x 0.5736 = 8.030..., so the opposite side is 8.0 cm (1 decimal place).
Practice questions
Try each question, then tap to reveal the answer.
Q1In a right-angled triangle, angle x = 40 degrees and the hypotenuse is 10 cm. Find the length of the side opposite x, to 1 decimal place.Show answer
Answer: 10 x sin(40) = 6.4278..., so 6.4 cm
Q2In a right-angled triangle, angle y = 55 degrees and the hypotenuse is 20 cm. Find the length of the side adjacent to y, to 1 decimal place.Show answer
Answer: 20 x cos(55) = 11.4715..., so 11.5 cm
Q3In a right-angled triangle, angle z = 28 degrees, and the side adjacent to z is 9 cm. Find the length of the side opposite z, to 1 decimal place.Show answer
Answer: 9 x tan(28) = 4.7854..., so 4.8 cm
Q4In a right-angled triangle, the side opposite angle p is 7 cm and the hypotenuse is 25 cm. Find angle p, to 1 decimal place.Show answer
Answer: p = sin^-1(7/25) = sin^-1(0.28) = 16.2602..., so 16.3 degrees
Q5In a right-angled triangle, the side adjacent to angle q is 15 cm and the hypotenuse is 17 cm. Find angle q, to 1 decimal place.Show answer
Answer: q = cos^-1(15/17) = 28.0724..., so 28.1 degrees
Q6In a right-angled triangle, the side opposite angle r equals the side adjacent to r, both 11 cm. Find angle r.Show answer
Answer: tan(r) = 11/11 = 1, so r = 45 degrees
Q7A right-angled triangle has a hypotenuse of 12 cm and an angle of 60 degrees. Find the length of the side opposite this angle, to 1 decimal place.Show answer
Answer: 12 x sin(60) = 10.3923..., so 10.4 cm
Q8In a right-angled triangle, relative to angle s, the opposite side is 18 cm and the adjacent side is 24 cm. Find angle s, to 1 decimal place.Show answer
Answer: s = tan^-1(18/24) = tan^-1(0.75) = 36.8698..., so 36.9 degrees
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A ramp for wheelchair access rises at an angle of 6 degrees to the horizontal. The ramp has a horizontal length of 8 m. Work out the vertical height the ramp rises, to 2 decimal places.
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A kite is flying on a taut string 45 m long. The string makes an angle of 52 degrees with the horizontal ground. (a) Work out the height of the kite above the ground, to 1 decimal place. (b) Work out the horizontal distance from the person holding the string to the point directly below the kite, to 1 decimal place.
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Free printable worksheet
Want more practice on paper? Download the introduction to trigonometry worksheet pack - 8 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 15 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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