Lesson 3 of 3Trigonometry of Acute Triangles

Unit 7 · Lesson 3 of 3

Choosing a Method and Solving Problems

Real problems don't arrive labelled. Sketch the triangle, mark what you know, and let the givens decide the tool: SOH CAH TOA for a right triangle, the sine law when you have an angle and its opposite side, the cosine law for SAS and SSS. Many problems chain two tools together.

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Key things to remember

1Right angle: SOH CAH TOA or the Pythagorean theorem. No right angle: the sine law or the cosine law.
2An angle and its opposite side known: sine law (AAS, ASA after the angle sum, SSA).
3Two sides and the angle between them (SAS) or three sides (SSS): cosine law.
4Multi-step problems: solve one triangle, then use its answer as a given in the next step or the next triangle.
5Check: angles add to $18 0^{\circ}$ , the longest side faces the largest angle, and sides to one decimal place and angles to the nearest degree are rounded only at the end.

A surveyor wants the width of a river without crossing it. A coast guard dispatcher needs the distance between two boats. A landscaper needs the angle at the corner of a triangular lot. Each of these is a triangle problem, but none of them says "use the sine law". The work in this lesson is deciding: turning words into a labelled sketch, spotting which measures you have, and picking the tool that can actually use them. Sometimes one tool finishes the job; often you need two in a row, where the answer from the first becomes a given for the second.

Sketch the triangle roughly to shape and label the vertices. Put each given measure in its place.
Mark the unknown with a question mark, and name the sides by their opposite angles.
Choose the tool from what is given (see the chart below). If two angles are known, find the third angle first.
Solve, keeping unrounded values in your calculator from one step to the next.
Check that the answer is reasonable, then state it in a sentence with units, sides to one decimal place and angles to the nearest degree.

Fig 1A flowchart for choosing a tool: a right angle means SOH CAH TOA or the Pythagorean theorem; two known angles give the third by the angle sum; an angle with its opposite side means the sine law; SAS means the cosine law for a side; SSS means the cosine law for an angle.

What you know	Name	First tool	Then
Two angles and a side not between them	AAS	Sine law	Angle sum, then the sine law for the last side
Two angles and the side between them	ASA	Angle sum (third angle)	Sine law for the other sides
Two sides and the angle between them	SAS	Cosine law for the third side	Sine law for an angle (opposite the shorter known side), then the angle sum
Three sides	SSS	Cosine law for the largest angle	Cosine law or sine law for a second angle, then the angle sum
Two sides and an angle opposite one of them	SSA	Sine law for an angle	Angle sum, then the sine law (in an acute triangle; the ambiguous case is Grade 11)

Which tool to start with in a triangle with no right angle, and how to finish solving it.

ASine law

Needs a complete opposite pair: an angle and the side across from it.
Finds a side: $a = \frac{b sin A}{sin B}$ .
Finds an angle with $sin^{- 1}$ ; correct in an acute triangle.
Cases: AAS, ASA (after the angle sum), SSA.

BCosine law

Needs SAS or SSS: no complete pair is required.
Finds a side: $a^{2} = b^{2} + c^{2} - 2 b c cos A$ .
Finds an angle with $cos^{- 1}$ : $cos A = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}$ .
Cases: SAS, SSS. More arithmetic, so watch the order of operations.

A surveying total station on a tripod set among shrubs above a sandy beach and a rocky shoreline with waves and a blue sky. — PhotoMeasuring angles from two points on a shoreline makes a triangle that the sine law or cosine law can solve, giving distances across water that no tape could reach.Cvstr, CC BY 4.0, via Wikimedia Commons

VideoA short guide to choosing the sine law or the cosine law from the measures you are given.Watch on YouTube

Problems that take more than one step

Here is a classic. Two stakes, $A$ and $B$ , are 150 m apart on one bank of a river. A tree $T$ stands on the far bank. The angle between $A B$ and the line of sight to the tree is $6 2^{\circ}$ at $A$ and $7 4^{\circ}$ at $B$ . How wide is the river? The width is the perpendicular distance from the tree to the line $A B$ , which is not a side of triangle $A BT$ . So the plan has two stages. First, solve the oblique triangle: $∠ T = 4 4^{\circ}$ , and the sine law gives $A T$ . Second, drop a perpendicular from $T$ to $A B$ . It makes a right triangle with hypotenuse $A T$ and angle $6 2^{\circ}$ at $A$ , so the width is $A T sin 6 2^{\circ}$ , straight from SOH CAH TOA. Working out the numbers gives a width of about 183.3 m (see the first worked example). Mixing the new laws with right-triangle trig like this is very common.

Which angle to find with the sine law

After using the cosine law for a side, you could find either remaining angle. Choose the angle opposite the shorter of the two sides you are comparing. It is smaller than the other remaining angle, so it can't be the largest angle in the triangle, and it is certainly acute. That makes

sin^{- 1}

safe even before you know the triangle is acute, a habit that will pay off in Grade 11.

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Practice mode: each right or acute triangle has one unknown. Pick the tool that finds it in one step, then enter the value. Solve mode lets you check a whole solution step by step.

VideoFour examples that decide which law to use and then combine them to solve the whole triangle.Watch on YouTube

Checking your answers

The three angles should add to $18 0^{\circ}$ (rounded angles may be off by $1^{\circ}$ ; the unrounded ones should not).
The longest side should face the largest angle, and the shortest side the smallest angle.
Each side must be shorter than the other two sides added together.
A second route should agree: if you found a side with the cosine law, the sine law ratio $\frac{a}{sin A}$ should match $\frac{b}{sin B}$ for the finished triangle.
A sketch to scale (or the triangle solver above) should look like your numbers.

Acute triangles in this course

Apart from the right-angle comparison in the cosine law lesson, every triangle you solve with the sine law or the cosine law in this unit is acute. If you solve an SSS triangle and the largest angle comes out over

9 0^{\circ}

, the cosine law still gives the right answer, because

cos^{- 1}

handles obtuse angles. But sine law work with obtuse angles, including the ambiguous case, is part of Grade 11.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

Stakes

A

and

B

are 150 m apart on one bank of a river. From

A

, the angle between

A B

and the line of sight to a tree

T

on the far bank is

6 2^{\circ}

; from

B

it is

7 4^{\circ}

. Find the width of the river, to one decimal place.

Example 2

Two boats leave a dock at the same time. Their paths make an angle of

4 8^{\circ}

. One travels at 18 km/h and the other at 24 km/h. How far apart are they after 1.5 h, to one decimal place?

Example 3

A triangular lot has sides 45 m, 52 m and 60 m. Find the largest angle at a corner of the lot, to the nearest degree.

Example 4

A trail runs 2.4 km from a parking lot

P

to a lookout

L

, then 3.1 km from

L

to a waterfall

W

. The angle between the two trails at

L

6 6^{\circ}

. Find the straight-line distance from

P

W

to one decimal place, and the angle at

P

to the nearest degree.

Where marks get lost

Reaching for the sine law when no complete opposite pair is known (SAS or SSS). Check for a pair before writing the equation.
Answering a "how wide" or "how high" question with a slanted side of the triangle. The width of a river or the height of an object is a perpendicular distance, which usually needs one more right-triangle step.
Using SOH CAH TOA in a triangle that has no right angle. The primary ratios only work in right triangles, or in right triangles you create with an altitude.
Copying a rounded answer from step 1 into step 2. Use the stored, unrounded value so the final rounding is the only one.
Not checking: a longest side facing a small angle, or angles that add to far more or less than $18 0^{\circ}$ , means a side was paired with the wrong angle.

Flashcards 7

AAS or ASA: which tool?

SAS: which tool?

SSS: which tool?

Right triangle: which tool?

+ 3 more

Practice

A surveyor measures two sides of a triangular field and the angle between them. Which tool finds the third side?

Fire towers

A

and

B

are 25 km apart. Smoke is seen at

F

. The angle between

A B

and the line of sight to the smoke is

5 8^{\circ}

A

and

6 6^{\circ}

B

. How far is the fire from tower

A

, to one decimal place?

Two hikers leave camp on straight paths that make an angle of

7 5^{\circ}

. One walks 4.2 km and the other 5.5 km. How far apart are they, to one decimal place?

A triangular sail has sides 3.0 m, 4.0 m and 4.5 m. Find its smallest angle, to the nearest degree.

Stakes

A

and

B

are 200 m apart on one bank of a river. The angle between

A B

and the line of sight to a rock

R

on the far bank is

7 0^{\circ}

A

and

5 5^{\circ}

B

. Find the width of the river (the perpendicular distance from

R

to line

A B

), to one decimal place.

Put the steps in order to find the width of a river from two stakes A and B on one bank and a tree T on the other.

1Round the width to one decimal place and state it with units.
2Use width = AT sin A in the right triangle formed by the perpendicular from T.
3Use the sine law to find AT.
4Measure AB and the angles at A and B, and sketch triangle ABT.
5Find angle T with the angle sum.

Match each situation with the tool to use first.

A ladder against a vertical wall: you know its length and its angle with the ground

A triangular lot with all three sides measured: find a corner angle

Two boats on paths 48° apart: you know how far each has gone

Two observers a known distance apart each measure the angle to a balloon

Which of these reported results must contain an error?

In acute

△ A BC

you know

∠ A = 5 0^{\circ}

∠ C = 6 0^{\circ}

and

b = 20

cm. What should you do first to find side

a

What is the three-letter name for the case where you know two angles and the side between them?

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