Grade 11 · MCR3U · Functions

MCR3U: Cosine Law and 3D Problems

The cosine law solves the triangles the sine law cannot start: two sides and the angle between them, or all three sides. In a 3D problem you split the figure into flat triangles and solve them one at a time, often with SOH CAH TOA in one and the cosine law in the next.

How It Works

The law. In any triangle, $a^2 = b^2 + c^2 - 2bc\cos A$ , where $A$ is the angle across from side $a$ . When $A = 90^\circ$ , $\cos A = 0$ and the law becomes the Pythagorean theorem, so think of the last term as the correction for a corner that is not square.

When to use it. Use it for SAS, two sides and the angle between them, to find the third side. Use it for SSS, all three sides, to find an angle. If you know a side and the angle across from it, the sine law is quicker.

Finding an angle. Rearrange first: $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$ . A negative value means $A$ is obtuse, and the inverse cosine gives that obtuse angle directly. Unlike the sine law, the cosine law has no ambiguous case. When you need all three angles, find the largest one first, across from the longest side.

3D problems. Draw the figure, then shade one triangle at a time. Start with a triangle where you already know enough, solve it for a side it shares with the next triangle, and carry that side across. A tower or a cliff meets the ground at a right angle, so it usually gives right triangles. The triangle on the ground between two observers usually needs the cosine law or the sine law.

Rounding. Keep full calculator values from one triangle to the next, and round only the final answer.

5 Worked Examples

Easy first, test level last. Try each one on paper, then open the solution.

Easy
In $\triangle ABC$ , $b = 8$ , $c = 11$ and $\angle A = 52^\circ$ . Find $a$ .
Show solution
1. Two sides and the angle between them: SAS, so use the cosine law.
2. $a^2 = 8^2 + 11^2 - 2(8)(11)\cos 52^\circ = 185 - 176\cos 52^\circ \approx 76.64$ .
3. $a \approx \sqrt{76.64} \approx 8.75$ .
Answer: $a \approx 8.75$
Easy
A triangle has sides $5$ , $7$ and $10$ . Find its largest angle.
Show solution
1. The largest angle is across from the longest side, $10$ .
2. $\cos C = \frac{5^2 + 7^2 - 10^2}{2(5)(7)} = \frac{-26}{70} \approx -0.3714$ .
3. $C = \cos^{-1}(-0.3714) \approx 111.8^\circ$ . The negative cosine means the angle is obtuse.
Answer: $C \approx 111.8^\circ$
Medium
Two hikers leave the same point on straight paths $65^\circ$ apart. One walks $4.2$ km and the other $6.5$ km. How far apart are they?
Show solution
1. The paths and the gap form a triangle with two sides and the angle between them.
2. $d^2 = 4.2^2 + 6.5^2 - 2(4.2)(6.5)\cos 65^\circ \approx 59.89 - 23.07 = 36.82$ .
3. $d \approx \sqrt{36.82} \approx 6.07$ .
Answer: About $6.07$ km apart.
Medium
A triangular garden has sides $14$ m, $18$ m and $23$ m. Find its smallest angle.
Show solution
1. The smallest angle is across from the shortest side, $14$ m.
2. $\cos A = \frac{18^2 + 23^2 - 14^2}{2(18)(23)} = \frac{657}{828} \approx 0.7935$ .
3. $A = \cos^{-1}(0.7935) \approx 37.5^\circ$ .
Answer: About $37.5^\circ$
Test level
A $30$ m flagpole stands at point $F$ on level ground. From point $A$ the angle of elevation of the top is $35^\circ$ , and from point $B$ it is $50^\circ$ . On the ground, $\angle AFB = 70^\circ$ . Find the distance $AB$ .
Show solution
1. The pole is vertical, so each sight line gives a right triangle: $\tan 35^\circ = \frac{30}{AF}$ and $\tan 50^\circ = \frac{30}{BF}$ .
2. $AF = \frac{30}{\tan 35^\circ} \approx 42.84$ m and $BF = \frac{30}{\tan 50^\circ} \approx 25.17$ m.
3. On the ground, $\triangle AFB$ has two sides and the angle between them: $AB^2 = AF^2 + BF^2 - 2(AF)(BF)\cos 70^\circ \approx 1731.6$ .
4. $AB \approx \sqrt{1731.6} \approx 41.61$ , using the unrounded $AF$ and $BF$ .
Answer: $AB \approx 41.61$ m

Learn It on Martian Lab

Martian Lab teaches this in the 3D Trignometry Problems lesson of its MCR3U Trigonometric Ratios unit. Watch it, then take the Trigonometric Ratios practice tests to see if it stuck.

Watch the 3D Trignometry Problems lesson free

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Free Trigonometric Ratios practice test, with solutions

More Trigonometric Ratios topics: Sine Law and the Ambiguous Case · Special Angles and the CAST Rule

More MCR3U topics: Inverse Functions · Transformations of Functions · Domain, Range and Function Notation · Simplifying Rational Expressions · Operations with Rational Expressions · Max/Min and Vertex Form · Linear-Quadratic Systems · Quadratic Word Problems · Simplifying Radicals · Exponent Laws and Rational Exponents · Graphing Exponential Functions · Exponential Growth and Decay · Transformations of Sine and Cosine · Sinusoidal Word Problems · Arithmetic Sequences and Series · Geometric Sequences and Series · Pascal's Triangle and the Binomial Theorem · Simple and Compound Interest · Annuities: Future and Present Value

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