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Choosing the ratio that fits what you know, and using it to find a missing side or a missing angle in context.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you turn the three trigonometric ratios into a method. You label the sides of a right triangle from the angle you are working with, choose the one ratio that links the side you know to the side you want, and decide whether that ratio has to be multiplied or divided by. You learn the exact ratios of $30^\circ$ and $45^\circ$ so that ramps, gates and ladders can be solved without a calculator, you use a known ratio to find an angle rather than a side, and you meet angles that are handed to you outside the triangle, such as an angle of depression, and move them inside it before you compute.
You can label the three sides of a right triangle — opposite, adjacent and hypotenuse — for a chosen acute angle, and you know that sine is opposite over hypotenuse, cosine adjacent over hypotenuse and tangent opposite over adjacent. You can also solve a one-step equation such as $\frac{3}{4} = \frac{h}{20}$. This lesson adds only the decision: which ratio, and multiply or divide.
Solving a triangle: finding every side and angle it has left unnamed, not just one of them.
Angle of elevation: the angle you look up through, measured from the horizontal. Angle of depression: the angle you look down through, from the horizontal.
Exact value: a value written as it truly is — $\frac{1}{2}$, $1$, $\frac{7}{25}$ — rather than rounded. The special angles $30^\circ$, $45^\circ$ and $60^\circ$ have exact ratios worth knowing by heart.
Benchmark: a ratio you know exactly and can compare an unknown ratio against, the way $\frac{1}{2}$ tells you whether an angle is above or below $30^\circ$.
Solving a right triangle is three questions in a fixed order. Which side do I want? Which side do I know? Which single ratio uses exactly those two? Opposite and hypotenuse means sine; adjacent and hypotenuse means cosine; opposite and adjacent means tangent. Then it is one equation. If the unknown sits on top, multiply: with $\tan\theta = \frac{3}{4}$ and $40$ m of ground, the height is $40 \times \frac{3}{4} = 30$ m. If the unknown sits underneath, divide: a rise of $30$ m with the same ratio gives $30 \div \frac{3}{4} = 40$ m. Three angles give exact ratios you should simply know: at $30^\circ$ the opposite side is exactly half the hypotenuse, so $\sin 30^\circ = \frac{1}{2}$ and a ramp rising $4$ m at $30^\circ$ is $8$ m long; at $45^\circ$ the other acute angle is $45^\circ$ too, so the legs are equal and $\tan 45^\circ = 1$; at $60^\circ$ the adjacent side is half the hypotenuse. Going the other way, a ratio finds an angle: equal legs give $\tan\theta = 1$, so $\theta = 45^\circ$, and a ratio smaller than $\frac{1}{2}$ means an angle smaller than $30^\circ$, because a bigger sine always belongs to a bigger angle.
Another way: steps
Mark the angle. Label the three sides from it: opposite, adjacent, hypotenuse. Circle the one you know and the one you want. Write the ratio that names both, put the numbers in, and solve the single equation left.
Another way: picture
A ladder against a wall: the ground angle marked, the wall as the opposite side, the ground as the adjacent side, the ladder as the hypotenuse. Any two of those three, plus the angle, give the third.
Choosing the ratio before naming the sides. Opposite and adjacent are not properties of a side; they depend on which angle you are working from. Mark the angle, label the three sides for that angle, and only then does one ratio stand out as the one using exactly the two sides in play.
Multiplying when you should divide. $\sin\theta = \frac{o}{h}$ is one equation with three letters. If the unknown is on top you multiply; if it is underneath you divide. A quick check catches it: the hypotenuse is the longest side, so an answer for it that came out smaller than a leg is upside down.
Forgetting that the angle may not be inside the triangle. An angle of depression is measured from a horizontal line at the top, and a bearing is measured from north. Neither is a corner of your triangle until you move it there — usually with alternate angles across a pair of parallel lines.
The height is opposite $\theta$ and the $40$ m is adjacent, so tangent is the ratio that uses exactly those two.
Name the sides before choosing.
$\frac{3}{4} = \frac{\text{height}}{40}$: the unknown is on top, so multiply.
Height $= 40 \times \frac{3}{4} = 30$ m.
Check: the mast is shorter than the distance, as a ratio below $1$ says it should be.
The $4$ m rise is opposite the $30^\circ$ angle and the ramp is the hypotenuse: sine.
At $30^\circ$ the opposite side is exactly half the hypotenuse, so $\sin 30^\circ = \frac{1}{2}$ and $\frac{1}{2} = \frac{4}{\text{ramp}}$.
The unknown is underneath, so divide.
Ramp $= 4 \div \frac{1}{2} = 8$ m.
It is longer than the rise, which every hypotenuse must be.
The height is opposite the angle and the string is the hypotenuse, so use sine.
$\sin 30^\circ = \frac{1}{2}$, and the unknown is on top, so multiply.
Height $= 30 \times \frac{1}{2} = 15$ m.
A ramp rises $3$ m to a doorway and makes an angle of $30^\circ$ with the level ground. How long is the sloping ramp itself, in metres?
answer
A gate has a diagonal brace that leaves the bottom rail at $45^\circ$ and runs up to the far upright. Along the rail it covers $15$ cm. How far up the upright does the brace reach, in centimetres?
Answer:
A surveyor stands $16$ m from the foot of a mast and measures the angle of elevation $\theta$ to its top. Her table gives $\tan\theta = \frac{7}{8}$. How tall is the mast, in metres?
Answer:
A right triangle has hypotenuse $12$ cm, and the side opposite the angle $\theta$ is $6$ cm. With no calculator and no table, what can you say about $\theta$?
A lighthouse keeper's eye is $36$ m above the sea, and she looks down at a boat. The angle of depression — the angle her line of sight makes below the horizontal — is $\theta$, and her chart gives $\tan\theta = \frac{4}{9}$. How far is the boat from the foot of the lighthouse, in metres?
Answer:
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A ramp rises $6$ m to a doorway and makes an angle of $30^\circ$ with the level ground. How long is the sloping ramp itself, in metres?
answer
You can pick the ratio that fits what you know and use it to find a missing side or a missing angle. Explain how you decide between multiplying and dividing by the ratio, and why the side opposite a $30^\circ$ angle is exactly half the hypotenuse.
8. Your turn: a kite string $30$ m long at $30^\circ$ to the ground — how high is the kite?, step 3