DoAssignment.ca

C2.3 · Choose and apply the sine law or cosine law in acute triangles

Learn to choose and apply the sine law or cosine law in acute triangles through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Geometry and Trigonometry

Grade 11 MBF3C — C2.3: Choose and apply a law in acute triangles

The sine law and cosine law help find missing measurements in triangles that are not necessarily right triangles. This lesson focuses on acute triangles. An acute triangle has three angles, each less than 90∘90^\circ. Your first decision is to look at the information given and choose the law that uses it. Before calculating, review how sides and angles are labelled: in a triangle labelled AA, BB, and CC, the lowercase side aa is opposite angle AA, and so on.

What you will learn

1. Review: label the triangle and its measurements

An angle is the opening formed by two sides. The side opposite an angle is directly across from it. For example, side aa is opposite angle AA. These matching pairs are important when using the sine law.
The angles inside any triangle add to 180∘180^\circ. If two angles are known, subtract their sum from 180∘180^\circ to find the third. An acute triangle has all three angles below 90∘90^\circ. A diagram may not be drawn to scale, so use the measurements provided rather than estimating from its appearance.
A+B+C=180∘A+B+C=180^\circ

2. Choose the law that fits the known information

The sine law compares opposite side-angle pairs. Choose it when you have a known pair, such as angle AA and opposite side aa, and enough other information to find the requested measurement. For example, two known angles and one side let you find another side. If necessary, first find the third angle using the triangle angle total.
The cosine law connects the three sides of a triangle with one of its angles. Use it to find a third side when you know two sides and the angle between them. That angle is called the included angle: it is where the two known sides meet. You can also use the cosine law to find an angle when all three sides are known.
A useful routine is to list the known sides and angles, identify any opposite side-angle pair, and check which law uses the information you have. Do not choose a law just because it looks familiar. Match the law to the measurements.
After calculating, check the result. Side lengths must be positive. Each angle in an acute triangle must be less than 90∘90^\circ, and all three angles must total 180∘180^\circ. Keep extra calculator digits during the calculation and round at the end.
asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}

3. Apply a law and check the result

When using the sine law, keep each side matched with its opposite angle. If you are finding an angle, use the inverse sine function on a calculator. Set the calculator to degrees because the angles in this lesson are measured in degrees.
When using the cosine law to find a missing side, the angle in the calculation must be between the two known sides. When finding an angle from three known sides, make sure the side opposite the target angle is in the matching position in the formula.
Keep side units consistent. If the given lengths are in metres, the calculated length is also in metres. A practical model might use two measured paths from one meeting point and the angle between them to find the direct distance between their endpoints. The calculated distance should fit the known lengths and angle.
c2=a2+b2−2abcos⁡Cc^2=a^2+b^2-2ab\cos C

Match the measurements to a law

Known informationUseful lawWhy it fits
An opposite side-angle pair and enough other informationSine lawIt compares opposite side-angle pairs.
Two sides and their included angle; find a third sideCosine lawIt uses the two sides and the angle between them.
All three sides; find an angleCosine lawIt relates the three sides to the angle opposite the target angle.

Worked example

Example 1: Find a side using the sine law

In an acute triangle, A=42∘A=42^\circ, B=68∘B=68^\circ, and side a=9.5a=9.5 cm. Find side bb to the nearest tenth of a centimetre.
  1. Identify the opposite pair
    Side aa is opposite angle AA, so the known opposite pair is 9.59.5 cm and 42∘42^\circ. The sine law relates this pair to side bb and its opposite angle, BB.
    asin⁡A=bsin⁡B\frac{a}{\sin A}=\frac{b}{\sin B}
  2. Substitute the measurements
    Replace the labels with their given values. Keeping the opposite pairs together gives an equation with only bb unknown.
    9.5sin⁡42∘=bsin⁡68∘\frac{9.5}{\sin 42^\circ}=\frac{b}{\sin 68^\circ}
  3. Solve for the side
    Multiply both sides by sin⁡68∘\sin 68^\circ to leave bb by itself. Evaluate the expression in degree mode and round the final result.
    b=9.5sin⁡68∘sin⁡42∘≈13.1 cmb=\frac{9.5\sin 68^\circ}{\sin 42^\circ}\approx 13.1\text{ cm}
Answer: Side bb is approximately 13.113.1 cm.
Check: Angle BB is larger than angle AA, so its opposite side should be longer than side aa. The third angle is 70∘70^\circ, making all three angles acute.

Worked example

Example 2: Find a side using the cosine law

Two sides of an acute triangle are 7.27.2 m and 10.010.0 m. Their included angle is 54∘54^\circ. Find the side opposite the 54∘54^\circ angle to the nearest tenth of a metre.
  1. Choose the law
    Two sides and the angle between them are known, so the cosine law directly gives the third side. Let cc be opposite the included angle, CC.
    c2=a2+b2−2abcos⁡Cc^2=a^2+b^2-2ab\cos C
  2. Substitute the measurements
    Use the two known lengths for aa and bb, and the included angle for CC.
    c2=7.22+10.02−2(7.2)(10.0)cos⁡54∘c^2=7.2^2+10.0^2-2(7.2)(10.0)\cos 54^\circ
  3. Calculate the side length
    Evaluate the expression in degree mode, then take the positive square root because a side length is positive.
    c=7.22+10.02−2(7.2)(10.0)cos⁡54∘≈8.3 mc=\sqrt{7.2^2+10.0^2-2(7.2)(10.0)\cos 54^\circ}\approx 8.3\text{ m}
Answer: The side opposite the 54∘54^\circ angle is approximately 8.38.3 m.
Check: The result is positive and shorter than either known side. This is reasonable for the given acute included angle.

Common mistakes and how to avoid them

Pairing an angle with a side beside it in the sine law.
Correction: Use the side directly opposite the angle. Side aa pairs with angle AA.
Using an angle that is not between the two known sides when finding a third side with the cosine law.
Correction: Use the included angle: the angle where the two known sides meet.
Using calculator settings that do not match angles measured in degrees.
Correction: Set the calculator to degrees before evaluating sine or cosine.
Rounding intermediate values too early.
Correction: Keep the calculator value through the calculation and round the final answer to the requested precision.

Lesson summary

Check your understanding

Question 1

An acute triangle has A=35∘A=35^\circ, B=75∘B=75^\circ, and a=8a=8 cm. Which law is the most direct choice for finding bb?
  1. Sine law
  2. Cosine law, because two sides are known
  3. Cosine law, because all three sides are known
  4. Triangle angle total alone
Show answer and explanation
Sine law
The known side aa is opposite the known angle AA. That opposite pair makes the sine law the direct choice.

Question 2

Two sides are 66 cm and 99 cm, and their included angle is 48∘48^\circ. Which law directly finds the third side?
  1. Sine law
  2. Cosine law
  3. Triangle angle total alone
  4. Opposite side-angle pairing alone
Show answer and explanation
Cosine law
Two sides and their included angle are the information needed to use the cosine law to find the third side.

Question 3

A triangle calculation gives angles of 41∘41^\circ, 63∘63^\circ, and 76∘76^\circ. Is the result consistent with an acute triangle?
  1. Yes; all three are less than 90∘90^\circ and total 180∘180^\circ.
  2. No; the angles total more than 180∘180^\circ.
  3. No; an acute triangle must have one angle equal to 90∘90^\circ.
  4. No; one angle is greater than 90∘90^\circ.
Show answer and explanation
Yes; all three are less than 90∘90^\circ and total 180∘180^\circ.
The angles total 180∘180^\circ, and each is less than 90∘90^\circ, so they are consistent with an acute triangle.

Key terms

Opposite side
The side directly across from a specified angle.
Included angle
The angle where two specified sides meet.
Acute triangle
A triangle with all three angles less than 90∘90^\circ.
Sine law
A rule that relates each side of a triangle to the sine of its opposite angle.
Cosine law
A rule that relates a side to the other sides and the angle between them.

Continue through MBF3C

View the complete Ontario Grade 11 Mathematics learning path

About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation C2.3. It is a study resource, not an official curriculum publication.

Official curriculum reference

Report a correction or ask a question