Chapter 6 — Trigonometry

Answer: A

Explanation

The right angle is at $C\mathclose{,}$ so the hypotenuse is $AB = 13$ — the side opposite the right angle.

For angle $A\mathclose{,}$ the opposite side is the one that does not touch $A\mathclose{:}$ $BC = 5\mathclose{.}$

$$\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{13}$$

The third side is $AC = \sqrt{13^2 - 5^2} = \sqrt{144} = 12\mathclose{,}$ which is where the other options come from. 5, 12, 13 is a Pythagorean triple worth memorising.

Why each wrong option is wrong:

• B, $\frac{5}{12}$ — $\frac{5}{12}$ is $\tan A\mathclose{:}$ opposite over adjacent.
• C, $\frac{12}{13}$ — $\frac{12}{13}$ is $\cos A\mathclose{:}$ adjacent over hypotenuse.
• D, $\frac{13}{5}$ — inverted the ratio. Sine and cosine are always at most 1, because the hypotenuse is the longest side, so any answer bigger than 1 is wrong immediately.

Takeaway: mark the angle, label opposite / adjacent / hypotenuse relative to it, then apply SOHCAHTOA. Sine and cosine can never exceed 1.

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Answer: G

Explanation

Sketch it: the ramp is the hypotenuse (9 m), the vertical rise is opposite the $12°$ angle, and the ground is adjacent.

Opposite and hypotenuse means sine:

$$\sin 12° = \frac{\text{height}}{9}$$

Multiply both sides by 9:

$$\text{height} = 9 \sin 12°$$

Sanity check without a calculator: $\sin 12°$ is a small number well below 1, so the height is a small fraction of 9 metres. That is what a gentle ramp should be.

Why each wrong option is wrong:

• F, $9\cos 12°$ — cosine gives the horizontal distance along the ground, not the rise.
• H, $\dfrac{9}{\sin 12°}$ — dividing by $\sin 12°$ gives about 43 metres, which is far longer than the ramp itself. Impossible: a leg cannot exceed the hypotenuse.
• J, $9\tan 12°$ — tangent uses the adjacent side, which is not given here.

Takeaway: pick the ratio that uses the side you have and the side you want. Then check the size: a leg is always shorter than the hypotenuse.

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Answer: D

Explanation

In a $30°$–$60°$–$90°$ triangle the sides are always in the ratio

$$1 : \sqrt{3} : 2$$

with the 1 opposite the $30°\mathclose{,}$ the $\sqrt{3}$ opposite the $60°\mathclose{,}$ and the 2 as the hypotenuse.

Here the side opposite $30°$ is 7, so that is the "1" of the ratio and everything scales by 7:

$$\text{hypotenuse} = 2 \times 7 = 14$$

The shortcut worth remembering: the hypotenuse is twice the shortest side, and the shortest side is always the one facing the smallest angle.

Why each wrong option is wrong:

• A, $3.5$ — halved rather than doubled. The hypotenuse is the longest side, so it cannot be smaller than a leg.
• B, $7 \sqrt{2}$ — used the $45°$–$45°$–$90°$ ratio $1 : 1 : \sqrt{2}\mathclose{.}$
• C, $7 \sqrt{3}$ — gave the side opposite $60°\mathclose{,}$ which is $7\sqrt{3}\mathclose{.}$

Takeaway: $30$–$60$–$90$ gives $1 : \sqrt{3} : 2\mathclose{,}$ and $45$–$45$–$90$ gives $1 : 1 : \sqrt{2}\mathclose{.}$ Match the side you are given to its place in the ratio, then scale.

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Answer: G

Explanation

The identity that links all three:

$$\tan \alpha = \frac{\sin \alpha}{\cos \alpha}$$

Rearrange it to make $\cos \alpha$ the subject:

$$\cos \alpha = \frac{\sin \alpha}{\tan \alpha} = \frac{5/13}{5/12} = \frac{5}{13} \times \frac{12}{5} = \frac{12}{13}$$

Check with the other identity, $\sin^2 + \cos^2 = 1\mathclose{:}$

$$\left(\frac{5}{13}\right)^2 + \left(\frac{12}{13}\right)^2 = \frac{25 + 144}{169} = \frac{169}{169} = 1 \; \checkmark$$

Why each wrong option is wrong:

• F, $\frac{13}{12}$ — divided tangent by sine instead of sine by tangent, giving $\frac{13}{12}\mathclose{.}$ A cosine can never exceed 1, so this is wrong on sight.
• J, $\frac{5}{13}$ and H, $\frac{5}{12}$ — repeated a quantity already given.

Takeaway: $\tan = \frac{\sin}{\cos}$ and $\sin^2 + \cos^2 = 1\mathclose{.}$ Between them they connect any two ratios. And sine and cosine are never above 1 — use that to reject an answer instantly.

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Section 2 — Beyond Right Triangles

When a triangle has no right angle, SOHCAHTOA does not apply. Two formulas cover every such case, and the ACT expects both from memory.

The law of sines — use it when you have an angle and the side opposite it:

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

The law of cosines — use it when you have two sides and the angle between them, or all three sides:

$$c^2 = a^2 + b^2 - 2ab\cos C$$

Notice that if $C = 90°$ then $\cos C = 0$ and the law of cosines collapses to $c^2 = a^2 + b^2$ — Pythagoras. It is the general version of a rule you already know.

Which to use: if you can pair a side with the angle opposite it, use sines. Otherwise use cosines.

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Answer: B

Explanation

Two sides and the angle between them: that is the law of cosines.

$$c^2 = a^2 + b^2 - 2ab\cos C$$

with $a = 7\mathclose{,}$ $b = 9$ and $C = 60°\mathclose{.}$ Since $\cos 60° = \frac{1}{2}\mathclose{:}$

$$c^2 = 49 + 81 - 2(7)(9)\left(\tfrac{1}{2}\right) = 130 - 63 = 67$$

$$c = \sqrt{67} \approx 8.19$$

Sanity check: the third side of a triangle must be less than the sum of the other two $(16)$ and more than their difference $(2)\mathclose{.}$ $8.19$ sits comfortably between. Correct.

Why each wrong option is wrong:

• A, $2$ — subtracted the sides. Like the sum, the difference is a limit the third side can approach and never reach.
• C, $\sqrt{130}$ — used Pythagoras, which assumes a right angle. The angle here is $60°\mathclose{,}$ so the $-2ab\cos C$ term cannot be dropped.
• D, $16$ — added the two sides, which is the absolute maximum a third side could approach and never reach.

Takeaway: two sides and the angle between them means the law of cosines. The $-2ab\cos C$ term is the correction that turns Pythagoras into a rule for any triangle.

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Answer: H

Explanation

Angles paired with their opposite sides: use the law of sines.

$$\frac{p}{\sin P} = \frac{q}{\sin Q}$$

$$\frac{p}{\sin 30°} = \frac{10}{\sin 45°}$$

$$p = \frac{10 \sin 30°}{\sin 45°} = \frac{10 \times \tfrac{1}{2}}{\tfrac{\sqrt{2}}{2}} = \frac{5}{\tfrac{\sqrt{2}}{2}} = \frac{10}{\sqrt{2}} = 5\sqrt{2} \approx 7.07$$

Sanity check: angle $P$ $(30°)$ is smaller than angle $Q$ $(45°)\mathclose{,}$ so the side opposite $P$ must be shorter than the side opposite $Q\mathclose{.}$ $7.07 < 10\mathclose{.}$ Correct.

Why each wrong option is wrong:

• F, $5$ — multiplied by $\sin 30°$ and never divided by $\sin 45°\mathclose{.}$
• G, $\frac{20}{3}$ — used the angles themselves in place of their sines, $\frac{p}{30} = \frac{10}{45}\mathclose{.}$ Sides are proportional to the sines of the opposite angles, not to the angles.
• J, $10 \sqrt{2}$ — inverted the ratio, giving about 14.1. That would make the side opposite the smaller angle the longer one, which the sanity check rejects.

Takeaway: the law of sines needs a side paired with its opposite angle. Then check the ordering: the bigger angle always faces the longer side.

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Section 3 — Radians and Graphs

A radian is another unit for angle, like measuring in metres instead of feet. The one conversion to know:

$$\pi \text{ radians} = 180°$$

So to go from degrees to radians, multiply by $\dfrac{\pi}{180}\mathclose{;}$ the other way, multiply by $\dfrac{180}{\pi}\mathclose{.}$ If the answer has a $\pi$ in it, it is in radians.

For graphs of $y = a \sin(bx)\mathclose{:}$

• the amplitude is $|a|$ — the height from the middle to a peak
• the period is $\dfrac{2\pi}{b}$ — how far along before the wave repeats

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Answer: B

Explanation

Multiply by $\dfrac{\pi}{180}\mathclose{:}$

$$40 \times \frac{\pi}{180} = \frac{40\pi}{180} = \frac{2\pi}{9}$$

Cancel carefully: $\frac{40}{180}$ divides top and bottom by 20 to give $\frac{2}{9}\mathclose{.}$

Sanity check: $\frac{2\pi}{9}$ is about $0.70$ radians. A right angle is $\frac{\pi}{2} \approx 1.57$ radians, and $40°$ is less than $90°\mathclose{,}$ so an answer below 1.57 is right.

Why each wrong option is wrong:

• A, $\frac{2}{9}$ — divided by 180 and lost the $\pi\mathclose{,}$ leaving a number in no unit at all.
• C, $\frac{9}{2 \pi}$ — used the conversion the other way round.
• D, $40 \pi$ — multiplied by $\pi$ without dividing by 180.

Takeaway: degrees to radians multiplies by $\frac{\pi}{180}\mathclose{.}$ The $\pi$ stays in the answer — if it has vanished, the conversion went the wrong way.

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Answer: F

Explanation

Both are read straight off the graph.

Amplitude is the distance from the middle of the wave to its top. The curve swings about $y = 0\mathclose{,}$ rises to 3 and falls to $-3\mathclose{,}$ so the amplitude is 3.

Period is the horizontal length of one complete wave: from a starting point, up, down, and back to the same height going the same way. The curve starts at $0°$ going up, and is next at zero going up at $180°\mathclose{.}$ The period is $180°$ — and the picture, $360°$ wide, holds two whole waves.

(That also gives the constants: $a = 3\mathclose{,}$ and $b = \dfrac{360°}{180°} = 2\mathclose{.})$

Why each wrong option is wrong:

• G, amplitude 3, period $360°$ — $360°$ is the width of the picture, not of one wave. Two waves fit into it.
• H, amplitude 6, period $180°$ — 6 is the full height, from the bottom at $-3$ to the top at 3. The amplitude is measured from the middle, so it is half of that.
• J, amplitude 3, period $90°$ — $90°$ is the distance from a top $(45°)$ to the next bottom $(135°)\mathclose{,}$ which is half a wave.

Takeaway: amplitude is middle to top, not bottom to top. Period is one whole wave — out and back to the same height, heading the same way.

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Answer: D

Explanation

$$\sin^2\theta + \cos^2\theta = 1$$

$$\cos^2\theta = 1 - \left(\tfrac{3}{5}\right)^2 = 1 - \tfrac{9}{25} = \tfrac{16}{25}$$

$$\cos\theta = \tfrac{4}{5}$$

The angle is acute, so the cosine is positive. This is the 3-4-5 triangle in disguise.

Why each wrong option is wrong:

• A, $\frac{2}{5}$ — subtracted the ratios without squaring them.
• B, $\frac{16}{25}$ — found $\cos^2\theta$ and stopped before taking the square root.
• C, $\frac{3}{4}$ — gave $\tan\theta = \frac{3}{4}\mathclose{.}$

Takeaway: $\sin^2 + \cos^2 = 1$ links the two directly. Square, subtract, root — and expect an answer no greater than 1.

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Answer: J

Explanation

$$\tan\beta = \frac{\sin\beta}{\cos\beta} = \frac{15/17}{8/17} = \frac{15}{17} \times \frac{17}{8} = \frac{15}{8}$$

The 17s cancel, which is why this is quicker than rebuilding the triangle.

Check with the other identity: $\left(\frac{15}{17}\right)^2 + \left(\frac{8}{17}\right)^2 = \frac{225 + 64}{289} = 1\mathclose{.}$ ✓

Why each wrong option is wrong:

• F, $\frac{8}{15}$ — divided cosine by sine, giving $\cot\beta\mathclose{.}$
• G, $\frac{15}{17}$ — repeated the sine.
• H, $\frac{23}{17}$ — added the two ratios.

Takeaway: $\tan = \frac{\sin}{\cos}\mathclose{,}$ in that order. Unlike sine and cosine, a tangent may exceed 1.

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Answer: B

Explanation

Opposite and adjacent means tangent:

$$\tan 35° = \frac{\text{height}}{40} \quad\Longrightarrow\quad \text{height} = 40 \tan 35°$$

Sanity check: $35°$ is less than $45°\mathclose{,}$ so the tangent is below 1 and the tower is shorter than the 40 metres of ground. That rules out D immediately.

Why each wrong option is wrong:

• A, $40\sin 35°$ and C, $40\cos 35°$ — sine and cosine both need the hypotenuse, which is not given here.
• D, $\dfrac{40}{\tan 35°}$ — dividing by the tangent gives about 57 metres, taller than the distance, which the $35°$ check rejects.

Takeaway: two legs and no hypotenuse means tangent. Below $45°$ the tangent is under 1; above it, over 1 — a free size check.

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Answer: H

Explanation

Start from the law of cosines with $c = 9$ the side opposite $C\mathclose{:}$

$$c^2 = a^2 + b^2 - 2ab\cos C$$

$$81 = 36 + 49 - 84\cos C$$

$$84\cos C = 85 - 81 = 4 \quad\Longrightarrow\quad \cos C = \frac{4}{84} = \frac{1}{21}$$

The cosine is small and positive, so $C$ is a little under $90°$ — which makes sense, since $9^2 = 81$ is only just less than $6^2 + 7^2 = 85\mathclose{.}$

Why each wrong option is wrong:

• F, $- \frac{1}{21}$ — subtracted in the wrong order, giving a negative cosine and so an obtuse angle. The check above shows the angle is acute.
• G, $0$ — a cosine of 0 would mean exactly $90°\mathclose{,}$ which needs $9^2 = 6^2 + 7^2\mathclose{.}$ It is not: 81 against 85.
• J, $\frac{2}{21}$ — divided by $ab = 42$ instead of $2ab = 84\mathclose{,}$ losing the 2.

Takeaway: the law of cosines works in both directions. Rearranged it gives $\cos C = \frac{a^2 + b^2 - c^2}{2ab}\mathclose{,}$ and the sign of the answer tells you whether the angle is acute or obtuse.

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Answer: A

Explanation

Sketch it. The ladder is the hypotenuse (10 m), the ground distance is adjacent to the $65°$ angle, and the wall is opposite.

Adjacent and hypotenuse means cosine:

$$\cos 65° = \frac{\text{distance}}{10} \quad\Longrightarrow\quad \text{distance} = 10\cos 65°$$

Since $65°$ is a steep angle, the cosine is small, so the foot of the ladder sits close to the wall — about 4.2 metres. That is what a steep ladder should do.

Why each wrong option is wrong:

• B, $10\sin 65°$ — sine gives the height reached up the wall, not the distance along the ground.
• C, $10\tan 65°$ — tangent needs the opposite and adjacent sides, and the hypotenuse is what is given.
• D, $\dfrac{10}{\cos 65°}$ — dividing gives about 24 metres, longer than the ladder. A leg cannot exceed the hypotenuse.

Takeaway: pick the ratio joining the side you have to the side you want. Then check the size against the hypotenuse.

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Answer: H

Explanation

Opposite over hypotenuse is the sine:

$$\sin\theta = \frac{7}{25}$$

To get $\theta$ itself, apply the inverse:

$$\theta = \sin^{-1}\!\left(\frac{7}{25}\right)$$

Why each wrong option is wrong:

• F, $\sin\left(\frac{7}{25}\right)$ — no inverse, so it computes the sine of a number rather than the angle whose sine is that number.
• G, $\sin^{-1}\left(\frac{25}{7}\right)$ — $\frac{25}{7}$ is greater than 1, and no angle has a sine above 1, so this expression has no value at all.
• J, $\cos^{-1}\left(\frac{7}{25}\right)$ — cosine uses the adjacent side, which here is 24, not 7.

Takeaway: to find a side use sin, cos or tan; to find an angle use their inverses. A sine or cosine above 1 is a signal you have inverted the ratio.

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Answer: C

Explanation

A $45$–$45$–$90$ triangle has sides in the ratio

$$1 : 1 : \sqrt{2}$$

with the $\sqrt{2}$ on the hypotenuse. Both legs are 6, so everything scales by 6:

$$\text{hypotenuse} = 6\sqrt{2} \approx 8.49$$

Check with Pythagoras: $6^2 + 6^2 = 72\mathclose{,}$ and $\sqrt{72} = 6\sqrt{2}\mathclose{.}$ ✓

Why each wrong option is wrong:

• A, $12$ — doubled the leg, which is the $30$–$60$–$90$ rule.
• B, $6 \sqrt{3}$ — used $\sqrt{3}\mathclose{,}$ also from the other special triangle.
• D, $3 \sqrt{2}$ — gave a hypotenuse shorter than the legs, which is impossible.

Takeaway: $45$–$45$–$90$ is $1 : 1 : \sqrt{2}\mathclose{;}$ $30$–$60$–$90$ is $1 : \sqrt{3} : 2\mathclose{.}$ Keep the two apart by remembering which has two equal sides.

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Answer: G

Explanation

Multiply by $\dfrac{180}{\pi}\mathclose{:}$

$$\frac{5\pi}{6} \times \frac{180}{\pi} = \frac{5 \times 180}{6} = 150°$$

The $\pi$ cancels, which is the sign the conversion is going the right way — an answer in degrees should have no $\pi$ left in it.

Sanity check: $\pi$ is $180°\mathclose{,}$ so $\frac{5\pi}{6}$ is a little less than that. 150 fits.

Why each wrong option is wrong:

• F, $30$ — divided 180 by 6 and forgot the 5.
• H, $216$ — inverted the fraction.
• J, $300$ — used $360°$ instead of $180°\mathclose{.}$

Takeaway: radians to degrees multiplies by $\frac{180}{\pi}\mathclose{,}$ and the $\pi$ must cancel. If a $\pi$ survives, the conversion went the wrong way.

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Answer: A

Explanation

For $y = a\cos(bx)$ the period is

$$\frac{2\pi}{b} = \frac{2\pi}{3}$$

The 5 outside sets the amplitude and has no effect on the period at all.

Note the direction: a bigger number inside gives a shorter period, because it squeezes the wave. Three full cycles now fit where one used to.

Why each wrong option is wrong:

• B, $3$ — gave the coefficient.
• C, $2 \pi$ — the period of $\cos x\mathclose{,}$ ignoring the coefficient.
• D, $6 \pi$ — multiplied by 3 rather than dividing, stretching the wave when it should be squeezed.

Takeaway: period is $\frac{2\pi}{b}\mathclose{,}$ where $b$ is the number multiplying $x\mathclose{.}$ Outside sets height, inside sets width — and inside works in reverse.

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Answer: F

Explanation

Read the highest and the lowest point of the curve: it reaches 24 at $90°$ and falls to 16 at $270°\mathclose{.}$

The midline is the level halfway between them — the level the wave swings about. It is what the constant added on the end sets:

$$d = \frac{24 + 16}{2} = 20$$

The amplitude is the distance from the midline to the top, and it is what the coefficient in front of the sine sets:

$$a = 24 - 20 = 4$$

Check: $4\sin(90°) + 20 = 24$ ✓ and $4\sin(270°) + 20 = 16$ ✓

Why each wrong option is wrong:

• G, $a = 8$ and $d = 20$ — 8 is the whole swing, from the bottom at 16 to the top at 24. The amplitude is measured from the midline, so it is half of that.
• H, $a = 4$ and $d = 24$ — 24 is the maximum, not the midline. The wave spends as much time above its midline as below it, and this curve is never above 24.
• J, $a = 20$ and $d = 4$ — the two constants exchanged. With $a = 20$ the curve would swing 20 either side of its midline, from $-16$ up to 24.

Takeaway: for $y = a\sin(x) + d\mathclose{,}$ the graph runs from $d - a$ to $d + a\mathclose{.}$ Halfway between top and bottom is $d\mathclose{;}$ from there to the top is $a\mathclose{.}$

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Answer: C

Explanation

$$\sin^2\theta + \cos^2\theta = 1$$

$$\sin^2\theta = 1 - \left(\tfrac{7}{25}\right)^2 = 1 - \tfrac{49}{625} = \tfrac{576}{625}$$

$$\sin\theta = \tfrac{24}{25}$$

The angle is acute, so the sine is positive. This is the 7-24-25 triangle.

Why each wrong option is wrong:

• A, $\frac{7}{24}$ — adjacent over opposite, $\frac{7}{24}\mathclose{.}$ That is neither the sine nor the tangent.
• B, $\frac{18}{25}$ — subtracted the ratio without squaring it.
• D, $\frac{25}{7}$ — inverted the cosine, giving a value above 1, which no sine can take.

Takeaway: square, subtract from 1, root. And use "no more than 1" to reject an answer before doing any arithmetic.

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Answer: F

Explanation

The fraction is the definition of the tangent:

$$\frac{\sin\theta}{\cos\theta} = \tan\theta$$

So the expression is $\tan\theta \times \cos\theta\mathclose{.}$ Writing the tangent back out shows the cancellation:

$$\frac{\sin\theta}{\cos\theta} \times \cos\theta = \sin\theta$$

Check at $\theta = 30°\mathclose{:}$ $\tan 30° \approx 0.577$ and $\cos 30° \approx 0.866\mathclose{,}$ and their product is $0.5 = \sin 30°\mathclose{.}$ ✓

Why each wrong option is wrong:

• G, $\cos\theta$ — cancelled the sine instead of the cosine.
• H, $\tan\theta$ — stopped at the tangent without multiplying.
• J, $1$ — cancelled everything, which would need the numerator and denominator to be identical.

Takeaway: $\tan = \frac{\sin}{\cos}$ works in both directions. When an expression mixes the three ratios, rewrite the tangent as a fraction and look for what cancels.

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Answer: C

Explanation

$$\sin^2\theta + \cos^2\theta = 1 \qquad \text{for every } \theta$$

So the expression is

$$1 + 3 = 4$$

no matter what $\theta$ is. That is what makes it an identity rather than an equation: it holds everywhere, so no value of $\theta$ needs to be found.

Why each wrong option is wrong:

• A, $1$ — used the identity correctly and then left the $+3$ off.
• B, $3$ — treated the two squared terms as zero.
• D, $5$ — counted each term as 1, giving $1 + 1 + 3\mathclose{.}$ The identity says their sum is 1, not each of them.

Takeaway: spot $\sin^2 + \cos^2$ and replace the pair with 1 immediately. It is the most useful identity on the paper.

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Answer: J

Explanation

Sketch it. The mast (12 m) is opposite the $40°$ angle, and the cable is the hypotenuse. Opposite over hypotenuse is the sine:

$$\sin 40° = \frac{12}{\text{cable}} \quad\Longrightarrow\quad \text{cable} = \frac{12}{\sin 40°}$$

Here the unknown is on the bottom, so the equation is rearranged by dividing rather than multiplying — the opposite of the usual case.

Sanity check: a hypotenuse is always the longest side, so the answer must exceed 12. Dividing by $\sin 40° \approx 0.64$ gives about 18.7 metres. ✓

Why each wrong option is wrong:

• F, $12\tan 40°$ — tangent does not involve the hypotenuse at all.
• G, $12\sin 40°$ — multiplying gives about 7.7 metres, shorter than the mast it is attached to. Impossible for a hypotenuse.
• H, $\dfrac{12}{\cos 40°}$ — cosine uses the adjacent side, which is the ground distance and is not given.

Takeaway: when the unknown is the hypotenuse, it ends up on the bottom of the ratio, so you divide. Check the answer is the longest side.

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