Practice right-triangle trigonometry, including sine, cosine, tangent, side ratios, and real-world angle applications. Trigonometry helps students understand motion, geometry, forces, and engineering problems.
Trigonometry begins with right triangles. A right triangle has one 90-degree angle. The longest side is called the hypotenuse. The other two sides are called legs.
Problem: A right triangle has legs 6 and 8. Find the hypotenuse.
Step 1: Use the Pythagorean theorem.
a² + b² = c²
Step 2: Substitute the leg lengths.
6² + 8² = c²
Step 3: Simplify.
36 + 64 = c²
100 = c²
Step 4: Take the square root.
c = 10
Answer: The hypotenuse is 10.
You are ready to move on when you can identify the hypotenuse and use the Pythagorean theorem to find missing side lengths.
Sine, cosine, and tangent compare the sides of a right triangle. These ratios depend on the angle you are using. The three basic ratios are: sin = opposite / hypotenuse, cos = adjacent / hypotenuse, and tan = opposite / adjacent.
Problem: A right triangle has opposite side = 3, adjacent side = 4, and hypotenuse = 5. Find sin, cos, and tan.
Step 1: Use the sine ratio.
sin = opposite / hypotenuse = 3 / 5
Step 2: Use the cosine ratio.
cos = adjacent / hypotenuse = 4 / 5
Step 3: Use the tangent ratio.
tan = opposite / adjacent = 3 / 4
Answer: sin = 3/5, cos = 4/5, tan = 3/4
You are ready to move on when you can choose the correct side ratio for sine, cosine, and tangent.
Trigonometry can help find a missing side of a right triangle when you know one angle and one side. Choose sine, cosine, or tangent based on which sides are involved.
Problem: A right triangle has angle θ = 30° and hypotenuse = 10. Find the opposite side.
Step 1: Choose the ratio that uses opposite and hypotenuse.
sin θ = opposite / hypotenuse
Step 2: Substitute the known values.
sin 30° = opposite / 10
Step 3: Use sin 30° = 0.5.
0.5 = opposite / 10
Step 4: Multiply both sides by 10.
opposite = 5
Answer: The opposite side is 5.
You are ready to move on when you can choose the correct trig ratio and solve for a missing side.
Angles can be measured in degrees or radians. Degrees are common in geometry, while radians are used heavily in calculus. The key conversion to remember is 180° = π radians.
Problem: Convert 90° to radians.
Step 1: Use the relationship 180° = π radians.
Step 2: 90° is half of 180°.
Step 3: Half of π is π/2.
Answer: 90° = π/2 radians.
You are ready to move on when you can use 180° = π radians to convert common angles.
Trigonometry is used to solve real-world problems involving height, distance, angles, slopes, ramps, ladders, forces, and motion. The key is to identify the right triangle, choose the correct ratio, and solve for the missing value.
Problem: A ladder is 10 ft long and makes a 60° angle with the ground. Find the height the ladder reaches on the wall. Use sin 60° ≈ 0.866.
Step 1: Identify the sides.
The ladder is the hypotenuse. The wall height is opposite the 60° angle.
Step 2: Choose the sine ratio.
sin θ = opposite / hypotenuse
Step 3: Substitute the known values.
sin 60° = height / 10
Step 4: Use sin 60° ≈ 0.866.
0.866 = height / 10
Step 5: Multiply both sides by 10.
height ≈ 8.66 ft
Answer: The ladder reaches about 8.66 ft up the wall.
You are ready to move on when you can identify the right triangle in a real situation and choose sine, cosine, or tangent to solve the problem.