Trig Basics
In a right triangle, if the length of the hypotenuse is 10 and the length of one leg is 8, what is the sine of the angle opposite the leg?
sin(θ) = 8/10 = 0.8
Solve for θ: cos(θ) = 0.8
θ = cos^(-1)(0.8) ≈ 36.87°
Simplify the expression: sin(θ)cos(θ)
sin(θ)cos(θ) = (1/2)(1/2) = 1/4
Graph the function y = cos(θ) on the interval [0, 2π].
Cosine function graph
A ladder leans against a wall forming a 60° angle with the ground. If the ladder is 15 feet long, how high is the top of the ladder from the ground?
Height = 15 * sin(60°) = 15 * (√3/2) = 15√3/2 ≈ 12.99 feet
Given that sin(θ) = 0.6 and θ is in Quadrant I, what is the cosine of θ?
cos(θ) = √(1 - sin^2(θ)) = √(1 - 0.6^2) = √(1 - 0.36) = √(0.64) = 0.8
Find all solutions to the equation: sin(2θ) = 1
2θ = π/2 + 2πn, where n is an integer
Simplify the expression: tan(θ) + cot(θ)
tan(θ) + cot(θ) = tan(θ) + 1/tan(θ) = tan^2(θ) + 1 = sec^2(θ)
Graph the function y = sin(θ) on the interval [-π, π].
Sine function graph
An airplane is flying at an altitude of 10,000 feet. If the angle of depression to a target on the ground is 30°, how far is the target from the airplane?
Distance = 10,000 * tan(30°) = 10,000 * (√3/3) = 10,000√3/3 ≈ 5773.5 feet
If sec(θ) = 2, what is the tangent of θ?
tan(θ) = 1/sec(θ) = 1/2 = 0.5
Solve for x: tan(x) = -1
x = tan^(-1)(-1) ≈ -45°
Use a Pythagorean identity to find the value of cos(θ) if sin(θ) = 3/5
cos(θ) = √(1 - sin^2(θ)) = √(1 - (3/5)^2) = √(1 - 9/25) = √(16/25) = 4/5
Graph the function y = tan(θ) on the interval [-π/2, π/2].
Tangent function graph
A ferris wheel has a diameter of 50 meters and completes one full rotation every 2 minutes. If you are at the top of the ferris wheel, how fast are you moving horizontally?
Horizontal speed = 50 * π * (1/2) * (1/2) = 25π ≈ 78.54 meters per minute
In a right triangle, if the length of one leg is 5 and the length of the hypotenuse is 13, what is the cosine of the angle opposite the leg?
cos(θ) = 5/13
Find the general solution for θ in the equation: cos(θ) = -0.5
θ = cos^(-1)(-0.5) ≈ ±120°
Prove the identity: sin^2(θ) + cos^2(θ) = 1
sin^2(θ) + cos^2(θ) = 1
Graph the function y = 2sin(θ) - 1 on the interval [0, 2π].
Shifted sine function graph
A ship leaves port and travels on a bearing of 120° for 3 hours at a speed of 25 knots. How far has the ship traveled from the port?
Distance = 3 * 25 = 75 nautical miles
Given that cot(θ) = -3/4 and θ is in Quadrant III, what is the secant of θ?
sec(θ) = 1/cot(θ) = 1/(-3/4) = -4/3
Solve for θ in the equation: 3sin(θ) + 2 = 0
θ = sin^(-1)((-2)/3) ≈ -41.81° or 180° + 41.81° ≈ 221.81°
Simplify the expression: (sec(θ) + tan(θ))(sec(θ) - tan(θ))
(sec^2(θ) - tan^2(θ)) = 1 - sin^2(θ) = cos^2(θ)
Graph the function y = sec(θ) on the interval [0, π/2].
Secant function graph
A surveyor stands 50 meters away from the base of a building and measures the angle of elevation to the top of the building as 60°. How tall is the building?
Height = 50 * tan(60°) = 50 * √3 ≈ 86.60 meters