Trigonometric Ratios in Right Triangles

sin 30 deg = ___ cos 30 deg = ___ tan 30 deg = ___ sin 45 deg = ___ cos 45 deg = ___ tan 45 deg = ___ sin 60 deg = ___ cos 60 deg = ___ tan 60 deg = ___

In triangle ABC, angle A = 30 deg, hypotenuse = 10 cm. Find the opposite side.

In triangle PQR, angle P = 45 deg, adjacent PQ = 8 cm. Find the hypotenuse PR.

In triangle XYZ, angle X = 60 deg, adjacent XY = 5 cm. Find the opposite side YZ.

Opposite = 6 cm, Hypotenuse = 12 cm -> use sin^-1:

Adjacent = 7 cm, Hypotenuse = 14 cm -> use cos^-1:

Opposite = 9 cm, Adjacent = 9 cm -> use tan^-1:

A ladder 6 m long leans against a wall making a 60 deg angle with the ground. How high up the wall does it reach? (To 2 decimal places.)

From the top of a 15 m cliff, the angle of depression to a boat is 30 deg. How far from the base of the cliff is the boat?

a = 3, b = 4: c =

a = 5, c = 13: b =

a = 7, b = 24: c =

sin theta = 0.707. theta =

cos theta = 0.5. theta =

tan theta = 1.732. theta =

In a right triangle, opposite = 8 cm, hypotenuse = 16 cm. Find angle theta.

From a point 30 m from the base of a building, the angle of elevation to the roof is 50 deg. How tall is the building? (To 2 decimal places.)

An aeroplane at 2000 m altitude spots a runway at an angle of depression of 10 deg. How far is the plane horizontally from the runway?

A ship travels 50 km due East, then 30 km due North. Draw a diagram. Calculate the straight-line distance from the starting point to the final position, and the angle from North (bearing) at which the ship finished.

In a right triangle, angle θ = 35° and hypotenuse = 10 cm. Find the opposite side.

In a right triangle, angle θ = 52° and adjacent = 8 m. Find the opposite side.

In a right triangle, angle θ = 20° and hypotenuse = 15 m. Find the adjacent side.

In a right triangle, angle θ = 60° and opposite = 12 cm. Find the hypotenuse.

A right triangle has opposite = 7 cm and hypotenuse = 10 cm. Find angle θ.

A right triangle has adjacent = 5 m and hypotenuse = 8 m. Find angle θ.

A right triangle has opposite = 4 cm and adjacent = 3 cm. Find angle θ.

A ladder 6 m long leans against a wall, reaching 5 m up. Find the angle the ladder makes with the ground.

From 40 m away, the angle of elevation to the top of a building is 58°. How tall is the building?

A bird is sitting at the top of a 12 m pole. From the bird's perspective, the angle of depression to a cat on the ground is 35°. How far is the cat from the base of the pole?

An aeroplane at altitude 3000 m spots a landing strip at an angle of depression of 12°. How far (horizontally) is the aeroplane from the strip?

A surveyor stands at point A and measures the angle of elevation to the top of a hill as 28°. She then walks 200 m closer to the hill (to point B) and measures the angle of elevation as 42°. Draw a diagram showing this situation.

Set up a system of equations using tan for both angles. Use the variable h for the height of the hill and d for the horizontal distance from B to the base of the hill.

Solve for h. (This type of problem is called 'triangulation' and is used in GPS and surveying.)

A right triangle has legs of 5 cm and 12 cm. (a) Use Pythagoras to find the hypotenuse. (b) Find both acute angles using trig.

A triangle has an angle of 40° and hypotenuse 10 cm. (a) Find both legs using trig. (b) Verify the legs satisfy Pythagoras' theorem.

Explain SOH CAH TOA in your own words, including when to use each ratio.

Describe how to find an unknown side given an angle and another side. Include a worked example.

Describe how to find an unknown angle given two sides. Include a worked example.

Give one real-world example where trigonometry is used, and explain which ratio would be needed.

Two buildings stand on level ground, 30 m apart. From the roof of the shorter building (10 m tall), the angle of elevation to the top of the taller building is 40°. Draw a diagram.

Calculate the height of the taller building. Show all working.

From the roof of the taller building, what is the angle of depression to the base of the shorter building?

Design a right triangle with sides 3, 4, 5. Calculate all three trig ratios for each acute angle.

Write a word problem (e.g. a ladder, building, or ramp) using your 3-4-5 triangle. Include the question and full solution.

Swap your problem with a partner (or parent) and solve each other's work.

A line rises 3 m for every 4 m horizontal. What is its gradient? What is its angle of inclination (to horizontal)?

A road has gradient 1/8. What angle does it make with the horizontal? Express to the nearest degree.

Explain in your own words why gradient and tan(angle) are the same thing.

A line passes through the origin with gradient m = 2. What angle does it make with the positive x-axis? (Hint: tan θ = gradient.)

A line makes a 50° angle with the positive x-axis. What is its gradient? Give a decimal answer to 2 decimal places.

If a line makes a 45° angle with the x-axis, what is its gradient? Why is this a special case?

A ski slope drops 400 m vertically over a horizontal distance of 600 m. What angle does the slope make with the horizontal?

A 10 m ladder must reach a window 8 m high. At what angle must it lean against the wall?

The shadow of a tree is 20 m when the sun is at 35° elevation. How tall is the tree?

In a right triangle with angle θ, hypotenuse c, and legs a and b. Write sin θ = a/c and cos θ = b/c. Rearrange each to express a and b in terms of c.

Using the identity sin²θ + cos²θ = 1, substitute your expressions for sin θ and cos θ. What do you get?

What have you just derived? Why is this significant?

A fire lookout tower stands on flat ground. An observer in the tower at 30 m above ground sees two fires. Fire A is at a horizontal distance of 500 m. Fire B is at a horizontal distance of 800 m, in the same direction. Draw a diagram showing both fires and the tower.

Calculate the angle of depression to each fire from the observer's position.

What is the angle between the observer's sightlines to the two fires?

If the observer walks 20 m higher in the tower, recalculate the angle of depression to Fire A.

List the 6 key trig skills from this worksheet (e.g. finding a side using sin, finding an angle using cos⁻¹, etc.).

Rate yourself on each skill: Confident / Developing / Need more practice. Explain one area where you need more practice.

Write one trig problem for each skill level you identified — one you can do confidently, one that challenges you.

A swimmer can see a buoy at a 25° angle of elevation from the waterline, at a horizontal distance of 200 m. How high is the buoy above the water? (Hint: buoys float.)

A ramp leads from ground level to a loading dock 1.2 m high. The ramp must not exceed an angle of 10°. What is the minimum horizontal length of the ramp?

A triangular garden bed has a right angle. One side is 6 m (north-south) and the other is 8 m (east-west). What angle does the hypotenuse fence make with the east-west side?

Research Hipparchus or Al-Biruni — two historical mathematicians who developed trigonometry. Write 3 key facts about their contributions.

Trigonometry was developed for astronomy and navigation. Name two modern applications that were not possible until trigonometry was developed.

The word 'trigonometry' comes from Greek. What do the Greek words mean? How does this connect to what you have learned?

Draw any triangle (not right-angled) with sides a, b, c and angles A, B, C. The Sine Rule states: a/sin A = b/sin B = c/sin C. Use the Sine Rule to find side a if A = 40°, B = 60°, b = 10 cm.

Explain why the Sine Rule works even without a right angle. (Hint: drop a perpendicular to create two right triangles.)

When would you need the Sine Rule instead of SOH CAH TOA?

Before this worksheet, what did you know about trigonometry? What was new?

Which trig skill took the most practice to understand? How did you overcome difficulties?

Write a real-world problem (from your life or interests) that could be solved using trig. Provide a full solution.

Rate your overall understanding of trig ratios 1–10 and explain the rating.

A basketball is launched at 45° to the horizontal. If the launch speed is 7 m/s, find the initial horizontal and vertical velocity components using trig.

A sprinter runs in a straight line from the starting blocks at 5° off-centre. After 100 m, how far sideways has she moved from the centre of the track?

A soccer player kicks the ball at 30° and it travels 20 m along the ground. How high does it reach at its peak? (Use the initial angle to estimate — assume symmetrical path.)

A right triangle has legs of 5 m and 12 m. Method 1: Use Pythagoras to find the hypotenuse, then find both acute angles using sin⁻¹.

Method 2: Use tan⁻¹ to find both acute angles directly from the two legs. Do both methods give the same angles?

Which method was more efficient? Why?

How is trigonometry related to Pythagoras' theorem? Write an explanation with an example.

How is trigonometry related to similar triangles? (Hint: think about why trig ratios depend only on the angle, not the size of the triangle.)

How could you use coordinate geometry and trigonometry together to find the angle a line segment makes with the x-axis?

Application 1: Engineering. Describe how an engineer would use trig when designing a bridge. What angles and lengths might they calculate?

Application 2: Navigation. Describe how a sailor uses trig and bearings to plot a course. Include a diagram.

Application 3: Your choice. Describe a trig application from a field you are interested in. Show at least one calculation.

If sin θ = 3/5, find cos θ using sin²θ + cos²θ = 1. Show all working.

If cos θ = 5/13, find sin θ and tan θ.

Explain why sin²θ + cos²θ = 1 is sometimes called the Pythagorean identity.

Explain, step by step, how you would solve for all unknown sides and angles in any right triangle given an angle and one side.

Create a right triangle problem at each of three difficulty levels: basic, intermediate, advanced. Provide full solutions for each.

If a friend had never heard of trigonometry, how would you introduce the idea in a way that connects to something they already know?

A surveyor needs to find the width of a river. From point A on one bank, she sights point B on the other bank at 90°. She walks 100 m along the bank to point C and sights point B at 62° from AC. Draw a diagram and calculate the width of the river (AB).

Show all your working and explain each step in words.

Write the three trig ratios (SOH CAH TOA) and draw a labelled right triangle showing all three sides.

Describe the process for finding (a) an unknown side and (b) an unknown angle. Use examples.

What is the most interesting trig application you encountered in this worksheet? Explain why.