Sine, cosine, tangent (intro)

An angle gives away the sides

The shadow method needs sun. What if it is not shining? There is a tool that calculates the sides of a triangle from an angle — and that does not need sun. It is called trigonometry.

It only works in a right-angled triangle. Its sides have names: the longest, opposite the right angle, is the hypotenuse. The other two are the legs — and according to the chosen angle they split into the opposite (across from the angle) and the adjacent (next to the angle).

And now the main thing. Thanks to similarity (the last lessons!) this holds: all right-angled triangles with the same angle have the same ratios of sides. A small and a giant triangle with an angle of 30° has the ratio of opposite to hypotenuse always 0.5.

Those ratios got names: sine (opposite/hypotenuse), cosine (adjacent/hypotenuse) and tangent (opposite/adjacent). A calculator knows their values under the buttons sin, cos, tan.

Today we will only meet them and calculate the first heights. No stress — this is an intro, not a test.

Jonáš stands under a lookout tower and aims at its top with an angle-measuring app. “I see the top at an angle of 35 degrees and I stand 40 metres from the base.” Sofie hunts for a calculator: “Tangent of 35° is about 0.7. Tangent is opposite over adjacent — height over distance. So the height is 40 times 0.7… 28 metres.” Jonáš glances at the tower leaflet: 28 metres including the viewpoint. “No shadow, no climbing. An angle and one distance are enough.”

Three ratios and how to calculate with them

Step 1: Sketch a right-angled triangle and label the sides with respect to the given angle α: hypotenuse (opposite the right angle), opposite (across from α), adjacent (next to α).

Step 2: Pick the right ratio according to what you know and what you look for.

sin α = opposite / hypotenuse — you know the hypotenuse, you look for the opposite (or the other way). cos α = adjacent / hypotenuse — the adjacent and the hypotenuse are in play. tan α = opposite / adjacent — both legs are in play (no hypotenuse anywhere).

Step 3: Find the value on a calculator. Check it is switched to degrees (DEG). A few values for a feel: sin 30° = 0.5. cos 60° = 0.5. tan 45° = 1 (angle 45° = both legs the same).

Step 4: Substitute and solve the equation. Example: hypotenuse 10 cm, angle 30°, I look for the opposite. Opposite and hypotenuse are in play → sine. sin 30° = x/10, so 0.5 = x/10, so x = 5 cm.

Step 5: Sense check. A leg must come out shorter than the hypotenuse. Always.

Why it works: two right-angled triangles with the same α are similar by AA — so they have the same ratios of sides. Trigonometry is packed similarity.

Example 1: the height of a slide (sine)

Question: A slide is 4 m long (that is the hypotenuse) and makes an angle of 30° with the ground. How high is its start?

Method: Height is across from the angle → opposite. I know the hypotenuse. Opposite and hypotenuse → sine.

sin 30° = height / 4.

Value: sin 30° = 0.5. So 0.5 = height/4, so height = 2 m.

Sense check: 2 m is less than the length of the slide 4 m — a leg shorter than the hypotenuse, it matches.

Answer: The start of the slide is 2 m above the ground.

Example 2: the height of a lookout tower (tangent)

Question: You stand 50 m from the base of a tower and see the top at an angle of 40°. How high is the tower? (tan 40° ≈ 0.84)

Method: Height of the tower = opposite, distance on the ground = adjacent. Both legs, no hypotenuse anywhere → tangent.

tan 40° = height / 50.

0.84 = height/50, so height = 50 · 0.84 = 42 m.

Sense check: 42 m is a tall tower, but real (the Petrin lookout tower in Prague is 63.5 m).

Answer: The tower is about 42 m high.

Example 3: a ladder against a wall (cosine)

Question: A ladder 5 m long leans against a wall at an angle of 60° from the ground. How far from the wall does its foot stand? (cos 60° = 0.5)

Method: The distance of the foot from the wall is next to the angle → adjacent. The ladder is the hypotenuse. Adjacent and hypotenuse → cosine.

cos 60° = distance / 5.

0.5 = distance/5, so distance = 2.5 m.

Sense check: a steep ladder (60°) stands close to the wall — 2.5 m out of 5 m length matches.

Answer: The foot of the ladder is 2.5 m from the wall.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Start with ten. When it goes well, add more.

Practise Pythagoras’ theorem — with trigonometry it makes an inseparable pair in a right-angled triangle. Write a whole number.

Practise the questions

On paper

A sketch for every problem. Label hypotenuse, opposite and adjacent, only then pick the ratio.

1. Sketch a right-angled triangle with angle α and label all three sides with respect to α. Then repeat the labels for the other acute angle — what swapped?

2. A toboggan rope is 8 m long and makes an angle of 30° with the ground. How high does it start? (sin 30° = 0.5)

3. You stand 30 m from a tree, you see the top at an angle of 45°. How high is the tree? (tan 45° = 1 — why does it come out so neatly?)

4. A ladder 6 m, angle with the ground 60°. Distance of the foot from the wall? (cos 60° = 0.5) Check that it came out shorter than the ladder.

Now you 💪

1. In a sketch you can find the hypotenuse, the opposite and the adjacent for a given angle.
2. You pick the ratio according to which two sides are in play.
3. You know why all triangles with the same angle have the same ratios (similarity AA).

What to take from this lesson

• Trigonometry calculates the sides of a right-angled triangle from an angle.
• sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.
• The ratios work thanks to similarity: the same angle = the same ratios, no matter how big the triangle.
• Pick the ratio according to the sides in play. No hypotenuse anywhere = tangent.
• The calculator must be in degree mode (DEG). A leg always comes out shorter than the hypotenuse.

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