Trigonometry of Acute Triangles: Test Review

Everything for the Unit 7 test on one page: the key facts from all 3 lessons, the mistakes that cost the most marks, and a practice test that points you back to the lessons you need.

Key facts

The Sine Law

In $△ A BC$ , side $a$ is opposite $∠ A$ , side $b$ is opposite $∠ B$ , side $c$ is opposite $∠ C$ .
Sine law: $\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}$ , or flipped, $\frac{sin A}{a} = \frac{sin B}{b} = \frac{sin C}{c}$ .
It comes from writing the altitude two ways, $h = b sin A = a sin B$ , and dividing by $sin A sin B$ .
Use it when you know an angle and its opposite side plus one more angle or side (AAS, ASA after the angle sum, or two sides and an angle opposite one of them).
To find an angle, solve for its sine, then use $sin^{- 1}$ in degree mode. In an acute triangle the result is the angle you want; the two-triangle ambiguous case is Grade 11.
The three angles add to $18 0^{\circ}$ : use this to find a third angle before or after the sine law.

The Cosine Law

Cosine law: $a^{2} = b^{2} + c^{2} - 2 b c cos A$ (and similarly $b^{2} = a^{2} + c^{2} - 2 a c cos B$ , $c^{2} = a^{2} + b^{2} - 2 ab cos C$ ).
For an angle: $cos A = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}$ , where $a$ is the side opposite $∠ A$ .
Use it for SAS (find the third side) and SSS (find an angle): the cases with no complete opposite pair.
It is developed from an altitude using the Pythagorean theorem and $x = b cos A$ ; when $∠ A = 9 0^{\circ}$ , $cos A = 0$ and it becomes $a^{2} = b^{2} + c^{2}$ .
If the longest side squared is less than the sum of the squares of the other two, the triangle is acute.

Choosing a Method and Solving Problems

Right angle: SOH CAH TOA or the Pythagorean theorem. No right angle: the sine law or the cosine law.
An angle and its opposite side known: sine law (AAS, ASA after the angle sum, SSA).
Two sides and the angle between them (SAS) or three sides (SSS): cosine law.
Multi-step problems: solve one triangle, then use its answer as a given in the next step or the next triangle.
Check: 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.

Where marks get lost

Pairing a side with the wrong angle. Side $a$ must go with $∠ A$ , the angle across from it, not an angle at one of its ends. Always name each side by its opposite angle before writing the equation. (The Sine Law)
Trying the sine law with ASA before finding the third angle. With $∠ A$ , $∠ B$ and side $c$ , no pair is complete until you compute $∠ C = 18 0^{\circ} - ∠ A - ∠ B$ . (The Sine Law)
Leaving the calculator in radian mode. $sin 35$ in radians is about $- 0.43$ ; in degrees it is about $0.57$ . A negative sine or a side longer than makes sense is the warning sign. (The Sine Law)
Forgetting the last step when finding an angle: $sin B = 0.74$ is not the answer. Press $sin^{- 1}$ to turn the sine into an angle. (The Sine Law)
Rounding the sine to two decimals partway through, then rounding again at the end, which can change the final tenth. Keep the full calculator value until the last line. (The Sine Law)
Subtracting before multiplying: computing $(b^{2} + c^{2} - 2 b c) cos A$ instead of $b^{2} + c^{2} - (2 b c cos A)$ . For the garden this gives $9 cos 7 0^{\circ} \approx 3.08$ and a fence of about 1.8 m instead of 15.7 m. (The Cosine Law)
Using an angle that is not between the two known sides. In $a^{2} = b^{2} + c^{2} - 2 b c cos A$ , the angle must be the one formed by sides $b$ and $c$ , which is the one opposite $a$ . (The Cosine Law)
Forgetting the square root at the end: $a^{2} \approx 245.9$ means $a \approx 15.7$ , not 245.9. (The Cosine Law)
Putting the wrong side in the subtracted spot when finding an angle. In $cos A = \frac{b ^{2} + c ^{2} - a ^{2}}{2 b c}$ , the subtracted side $a$ is the one opposite the angle you want. (The Cosine Law)
Rounding each angle and then using rounded values in the angle sum. Rounded angles can add to $17 9^{\circ}$ or $18 1^{\circ}$ ; keep the unrounded values until the end. (The Cosine Law)
Reaching for the sine law when no complete opposite pair is known (SAS or SSS). Check for a pair before writing the equation. (Choosing a Method and Solving Problems)
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. (Choosing a Method and Solving Problems)
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. (Choosing a Method and Solving Problems)
Copying a rounded answer from step 1 into step 2. Use the stored, unrounded value so the final rounding is the only one. (Choosing a Method and Solving Problems)
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. (Choosing a Method and Solving Problems)

Practice test

18 questions from across the unit. Each one shows the lesson it comes from.

The Sine Law
1
In $△ A BC$ , which side is opposite $∠ B$ ?
The Sine Law
2
In acute $△ A BC$ , $∠ A = 7 0^{\circ}$ , $a = 16$ m, and $c = 13$ m. Find $∠ C$ , to the nearest degree.
°
The Sine Law
3
In which situation can you use the sine law straight away to find side $b$ , with no other step first?
The Sine Law
4
Put the steps of the sine law's development in order.
1. 1Write h two ways using the sine ratio: h = b sin A in one right triangle and h = a sin B in the other.
2. 2Divide both sides by sin A sin B to get a / sin A = b / sin B.
3. 3Since both expressions equal h, set them equal: b sin A = a sin B.
4. 4Draw the altitude h from C, at right angles to side AB, splitting the triangle into two right triangles.
The Sine Law
5
Which statement about using $sin^{- 1}$ with the sine law is correct?
The Sine Law
6
In $△ A BC$ , $∠ A = 3 5^{\circ}$ and $∠ B = 6 5^{\circ}$ . What is $∠ C$ , in degrees?
°
The Cosine Law
7
In $△ A BC$ , $a = 8$ m, $b = 11$ m, and $∠ C = 5 8^{\circ}$ . Find side $c$ , to one decimal place.
m
The Cosine Law
8
In $△ A BC$ , $a = 7$ , $b = 5$ and $c = 6$ . What is the exact value of $cos A$ ? (Give a decimal or a fraction.)
The Cosine Law
9
Which equation correctly finds side $b$ in $△ A BC$ ?
The Cosine Law
10
What is the value of $cos 9 0^{\circ}$ ? (This is why the cosine law turns into the Pythagorean theorem for a right angle.)
The Cosine Law
11
Which set of side lengths makes an acute triangle?
The Cosine Law
12
Match what you know about a triangle with the tool you use first.
Two sides and the angle between them
All three sides
Two angles and the side between them
A right angle and both legs
Choosing a Method and Solving Problems
13
A surveyor measures two sides of a triangular field and the angle between them. Which tool finds the third side?
Choosing a Method and Solving Problems
14
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?
km
Choosing a Method and Solving Problems
15
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 $7 0^{\circ}$ at $A$ and $5 5^{\circ}$ at $B$ . Find the width of the river (the perpendicular distance from $R$ to line $A B$ ), to one decimal place.
m
Choosing a Method and Solving Problems
16
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.
1. 1Round the width to one decimal place and state it with units.
2. 2Use width = AT sin A in the right triangle formed by the perpendicular from T.
3. 3Use the sine law to find AT.
4. 4Measure AB and the angles at A and B, and sketch triangle ABT.
5. 5Find angle T with the angle sum.
Choosing a Method and Solving Problems
17
Which of these reported results must contain an error?
Choosing a Method and Solving Problems
18
What is the three-letter name for the case where you know two angles and the side between them?

Endless drill

Fresh numbers every time, for the calculation questions in this unit.

Endless drillnew numbers every time

Generating a problem…