SL 3.2 Apply right-triangle trigonometry, sine rule, cosine rule, and triangle area | International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Geometry and Trigonometry

Choosing and applying the right method for a triangle

Triangles can be described by their side lengths, their angles, or both. The useful method depends on which information is known. This lesson reviews right-triangle trigonometry, then develops the sine rule, cosine rule, and area formula for triangles that need not contain a right angle. In each case, label the diagram before substituting values, keep angle units consistent, and round only at the end.

What you will learn

Use sine, cosine, and tangent to find unknown sides or angles in right-angled triangles.
Choose and apply the sine rule or cosine rule in non-right-angled triangles.
Find a triangle’s area using two sides and their included angle.
Use diagrams, units, and calculator results to check whether an answer is reasonable.

1. Prior knowledge: right triangles and labels

A right-angled triangle has one angle equal to

90^\circ

. Its hypotenuse is the side opposite that angle and is the longest side. Relative to a chosen acute angle

\theta

, the other two sides are called the opposite and adjacent sides. These names depend on which angle is chosen.

The three basic ratios are sine, cosine, and tangent. The calculator should be in degree mode when angles are given in degrees. For example, if the opposite side and hypotenuse are known, use sine; if the adjacent side and hypotenuse are known, use cosine; if the opposite and adjacent sides are known, use tangent.

In any triangle, use matching labels: angles

A

,

B

, and

C

are opposite sides

a

,

b

, and

c

, respectively. The lowercase side is always opposite the uppercase angle with the same letter. This convention makes the rules easier to apply.

\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad \tan\theta=\frac{\text{opposite}}{\text{adjacent}}

Check which sides are opposite and adjacent to the angle of interest.
Use degree mode for degree measures.
Label each side opposite its matching angle.

2. Right-triangle trigonometry

Select the ratio containing the known side and the unknown. If an angle is unknown, rearrange the ratio and use the matching inverse function on the calculator. For instance, an equation of the form

\sin\theta=x

gives

\theta=\sin^{-1}(x)

. Here,

\sin^{-1}

means inverse sine, not the reciprocal of sine.

The Pythagorean theorem is also useful when the triangle is right-angled: the squares of the two shorter sides add to the square of the hypotenuse. It can find a missing side without an angle. A sketch helps distinguish the hypotenuse before using the formula.

A diagram is the geometric representation; a ratio such as

\sin\theta=\frac{u}{v}

is the symbolic representation; a calculator gives a numerical value. In a context, include units for lengths and state the angle as a degree measure. Technology is useful for evaluating inverse trigonometric functions and checking arithmetic, but the chosen ratio must still be justified.

x^2+y^2=z^2

Choose a ratio from the sides available, not by guessing.
Use inverse trigonometric functions to find an angle.
The hypotenuse is opposite the right angle.

3. Non-right triangles: sine rule and cosine rule

The sine rule connects each side to the sine of its opposite angle. It is especially useful when you know an opposite side-angle pair and another angle or side. Set up corresponding pairs carefully:

a

must be paired with

A

, not with another angle.

The cosine rule relates all three sides to one angle. Use it when two sides and their included angle are known, or when all three sides are known and an angle is required. The included angle is the angle between the two known sides.

For a missing angle from three sides, rearrange the cosine rule before using inverse cosine. For a missing side, substitute the known included angle directly. A calculator can evaluate the final trigonometric value, while the labelled diagram and equation show why that calculation applies.

With the sine rule, an inverse-sine calculation may produce an angle that needs checking against the diagram and the triangle angle sum. A triangle’s interior angles add to

180^\circ

. Confirm that the resulting angles are consistent with the given information before accepting an answer.

\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C},\quad a^2=b^2+c^2-2bc\cos A

Use the sine rule when an opposite side-angle pair is available.
Use the cosine rule for two sides with their included angle, or for three known sides.
Check that angles in a triangle sum to $180^\circ$ .

4. Area, technology, and answer checks

When two sides and the angle between them are known, the area is half the product of those sides multiplied by the sine of the included angle. The angle must lie between the two sides used. This formula works for an oblique triangle as well as a right triangle.

Area is measured in square units. For example, if lengths are in centimetres, the area is in square centimetres. The numerical value of the area should be positive. If a question asks for a specified accuracy, retain calculator precision during intermediate steps and round only the final result.

Use graphing or scientific calculator features purposefully: select degree mode, evaluate trigonometric ratios or inverse functions, and keep enough displayed digits while working. A calculator check does not replace selecting the correct rule. Compare the result with the sketch: a side should be plausible relative to the other sides, and an angle should fit the triangle’s shape.

K=\frac{1}{2}ab\sin C

Use the included angle in the area formula.
Area has squared units.
State the final rounding and check the result against the diagram.

Worked example

Right-triangle ratios

A right-angled triangle has hypotenuse

25

cm and one leg

7

cm. Find the acute angle opposite the

7

cm side, to the nearest tenth of a degree.

Choose a ratio
Relative to the required angle, the known $7$ cm side is opposite and the $25$ cm side is the hypotenuse, so sine uses both known lengths.
$\sin\theta=\frac{7}{25}$
Find the angle
Use inverse sine in degree mode. The ratio is less than $1$ , as expected for opposite divided by hypotenuse.
$\theta=\sin^{-1}\left(\frac{7}{25}\right)\approx16.2602^\circ$
Round as requested
Rounding to the nearest tenth gives the required angle measure.
$\theta\approx16.3^\circ$

Answer: The angle is approximately

16.3^\circ

.

Check: The angle is acute, as it must be, and the opposite side is shorter than the hypotenuse.

Worked example

Sine rule with two known angles

In a triangle,

A=42^\circ

,

B=68^\circ

, and

a=9

cm. Find

b

and

c

, to three significant figures.

Find the third angle
The angles in a triangle sum to $180^\circ$ , so first find the angle opposite side $c$ .
$C=180^\circ-42^\circ-68^\circ=70^\circ$
Find side b
Side $a$ is paired with angle $A$ . Apply the same pairing to $b$ and $B$ in the sine rule.
$b=9\frac{\sin68^\circ}{\sin42^\circ}\approx12.5\text{ cm}$
Find side c
Use the known pair $a$ and $A$ again, now pairing $c$ with $C$ .
$c=9\frac{\sin70^\circ}{\sin42^\circ}\approx12.6\text{ cm}$

Answer: To three significant figures,

b=12.5

cm and

c=12.6

cm.

Check: The largest angle is

C=70^\circ

, so its opposite side

c

should be the longest. The values satisfy

c>b>a

.

Worked example

Cosine rule and area

Two sides of a triangle are

8

cm and

11

cm, and their included angle is

57^\circ

. Find the third side and the area, each to three significant figures.

Find the third side
The known angle lies between the sides of lengths $8$ and $11$ , so use the cosine rule with those sides adjacent to the angle.
$a=\sqrt{8^2+11^2-2(8)(11)\cos57^\circ}\approx9.44\text{ cm}$
Find the area
The two known sides and their included angle fit the triangle area formula directly.
$K=\frac{1}{2}(8)(11)\sin57^\circ\approx36.9\text{ cm}^2$

Answer: The third side is approximately

9.44

cm, and the area is approximately

36.9

cm

^2

.

Check: The third side is shorter than

11

cm, which is reasonable because the included angle is less than

90^\circ

. The area is positive and has square units.

Common mistakes and how to avoid them

Pairing a side with an angle that is not opposite it.

Correction: Label the diagram first; the sine rule pairs

a

with

A

,

b

with

B

, and

c

with

C

.

Using the cosine rule with an angle that is not between the two substituted sides.

Correction: Identify the included angle and use the two sides that meet at that angle.

Using a right-triangle ratio in a triangle that has no right angle.

Correction: For a non-right triangle, choose the sine rule, cosine rule, or area formula according to the known information.

Rounding intermediate values too early or omitting units.

Correction: Keep adequate calculator precision, round at the end, and give length or area units as appropriate.

Lesson summary

In a right triangle, choose sine, cosine, or tangent according to the sides known and required.
Use the sine rule with corresponding opposite side-angle pairs.
Use the cosine rule when two sides and their included angle, or all three sides, are known.
Use half the product of two sides and the sine of their included angle to find area.
Label, calculate, include units, and check that the result fits the triangle.

Check your understanding

Question 1

In a right triangle, an angle has opposite side

5

and hypotenuse

13

. Which equation finds the angle

\theta

?

$\theta=\sin^{-1}(5/13)$
$\theta=\cos^{-1}(5/13)$
$\theta=\tan^{-1}(5/13)$
$\theta=\sin(5/13)$

Show answer and explanation

\theta=\sin^{-1}(5/13)

Sine is opposite divided by hypotenuse, so the angle is found with inverse sine.

Question 2

In a triangle,

A=50^\circ

,

B=60^\circ

, and

a=10

cm. What is

C

?

$70^\circ$
$80^\circ$
$110^\circ$
$120^\circ$

Show answer and explanation

70^\circ

The interior angles sum to

180^\circ

, giving

C=180^\circ-50^\circ-60^\circ=70^\circ

.

Question 3

Two sides are

6

cm and

9

cm, and their included angle is

40^\circ

. Which expression gives the area?

$\frac{1}{2}(6)(9)\sin40^\circ$
$\frac{1}{2}(6)(9)\cos40^\circ$
$\frac{1}{2}(6+9)\sin40^\circ$
$\frac{1}{2}(6)(9)\tan40^\circ$

Show answer and explanation

\frac{1}{2}(6)(9)\sin40^\circ

The area formula uses half the product of the two sides and the sine of the included angle.

Key terms

Hypotenuse: The side opposite the right angle in a right-angled triangle.
Included angle: The angle between two specified sides.
Inverse trigonometric function: A calculator function, such as inverse sine, used to find an angle from a trigonometric ratio.

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SL 3.2 · Apply right-triangle trigonometry, sine rule, cosine rule, and triangle area

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Choosing and applying the right method for a triangle

What you will learn

1. Prior knowledge: right triangles and labels

2. Right-triangle trigonometry

3. Non-right triangles: sine rule and cosine rule

4. Area, technology, and answer checks

Right-triangle ratios

Sine rule with two known angles

Cosine rule and area

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through IB AA SL

About this lesson and its review