Simple Trig Equations
Consider sin(x) = 0.6.
(a) How many solutions are on [0, 2pi)?
(b) Which quadrants contain solutions?
(a) 2 solutions.
(b) Quadrants I and II (sine is positive there).
Solve on [0, 2pi): 4cos(theta) − 1 = 2.
(a) Rewrite as cos(theta) = ___.
(b) How many solutions on [0, 2pi)?
(a) 4cos(theta) = 3 → cos(theta) = 3/4
(b) 2 solutions (Quadrants I and IV).
Find the exact value in radians: sin⁻¹(1/2)
pi/6
The point (3, 4) lies on the terminal ray of angle theta in standard position.
(a) Find r.
(b) Find sin(theta), cos(theta), tan(theta) (exact).
(a) r = sqrt(3^2+4^2)=5
(b) sin(theta)=4/5, cos(theta)=3/5, tan(theta)=4/3
Ferris wheel radius 40 ft. Rider starts at 3 o’clock and rotates counterclockwise.
(a) Write an equation relating vertical position above the center, h, to angle theta.
(b) State the domain for h.
(a) h = 40sin(theta)
(b) -40 <= h <= 40
Solve on [0, 2pi): cos(theta) = -0.3.
(a) Use cos⁻¹ to find one solution.
(b) Find the other solution.
(c) Round to 2 decimals.
(a) theta1 = cos⁻¹(-0.3) ≈ 1.88
(b) theta2 = 2pi − theta1 ≈ 2pi − 1.88 ≈ 4.40
(c) Solutions: theta ≈ 1.88, 4.40
Solve on [0, 2pi): 3sin(x) + 5 = 6.
(a) Isolate sin(x).
(b) Find all solutions on [0, 2pi), round to 2 decimals.
(a) 3sin(x)=1 → sin(x)=1/3
(b) x1 = sin⁻¹(1/3) ≈ 0.34
x2 = pi − 0.34 ≈ 2.80
So x ≈ 0.34, 2.80
Find the exact value in radians: cos⁻¹(0)
pi/2
The point (-3, 4) lies on a circle of radius 5.
(a) Find theta (radians) measured counterclockwise from +x-axis (calculator allowed).
(b) State the quadrant and why.
(a) Reference angle = tan⁻¹(|4/3|) ≈ tan⁻¹(1.333...) ≈ 0.93
Quadrant II → theta ≈ pi − 0.93 ≈ 2.21
(b) Quadrant II because x is negative and y is positive.
A person stands 50 ft from the base of a flagpole on level ground.
The angle of elevation to the top of the pole is 35 degrees.
(a) Write a trigonometric equation to find the height of the pole.
(b) Identify which trig function is used and why.
(a) tan(35) = h / 50
(b) Tangent, because height is opposite and distance is adjacent.
Solve on [0, 2pi): tan(theta) = 2.4.
(a) How many solutions are on [0, 2pi)?
(b) Find both solutions (calculator allowed).
(c) Round to 2 decimals.
(a) 2 solutions.
(b) theta1 = tan⁻¹(2.4) ≈ 1.18
theta2 = theta1 + pi ≈ 1.18 + 3.14 ≈ 4.32
(c) Solutions: theta ≈ 1.18, 4.32
Solve on [0, 2pi): 2sin(theta) − 0.4 = 0.
(a) Isolate sin(theta).
(b) Find all solutions on [0, 2pi), round to 2 decimals.
(a) 2sin(theta)=0.4 → sin(theta)=0.2
(b) theta1 = sin⁻¹(0.2) ≈ 0.20
theta2 = pi − 0.20 ≈ 2.94
So theta ≈ 0.20, 2.94
Find the exact value in radians: tan⁻¹(1/sqrt(3))
pi/6
Suppose cos(theta)=0.73 and x=8 for a point (x,y) on a circle centered at the origin.
(a) Find r.
(b) Find y (both possible values).
(c) Which quadrants are possible?
(a) cos(theta)=x/r → 0.73
=8/r → r=8/0.73 ≈ 10.96
(b) Use r^2=x^2+y^2 → y=±sqrt(r^2−x^2)
=±sqrt(10.96^2−64)≈±7.49
(c) Quadrants I or IV (cos positive means x positive; y can be + or -).
A ladder is 15 ft long and leans against a wall, making an angle of 63 degrees with the ground.
(a) Write a trigonometric equation to find how high the ladder reaches on the wall.
(b) Find the height, rounded to one decimal place.
(a) sin(63) = h / 15
(b) h ≈ 15 sin(63) ≈ 13.4 ft
Consider sin(x) = -2/3.
(a) Does x represent an angle or a proportion?
(b) How many solutions are on [0, 2pi)?
(c) Approximate both solutions on [0, 2pi), round to 2 decimals.
(a) Angle.
(b) 2 solutions.
(c) One solution from sin⁻¹(-2/3) ≈ -0.73.
Convert to a positive angle: x1 ≈ 2pi − 0.73 ≈ 5.55 (Quadrant IV).
Other solution: x2 ≈ pi − (-0.73) = pi + 0.73 ≈ 3.87 (Quadrant III).
So x ≈ 3.87, 5.55
Consider 2cos(theta) − 1 = 0.06.
(a) Rewrite as cos(theta)=___.
(b) How many solutions are on [0, 2pi)?
(c) Which quadrants contain solutions?
(a) 2cos(theta)=1.06 → cos(theta)=0.53
(b) 2 solutions
(c) Quadrants I and IV (cosine positive).
Find the exact value in radians: sin⁻¹(-sqrt(3)/2)
-pi/3
A vector v = <-5, -9> is in standard position.
(a) Find ||v||.
(b) Find direction angle theta on [0, 2pi) (radians), round to 2 decimals.
(c) Explain why tan⁻¹(9/5) alone is not the final answer.
(a) ||v||=sqrt(25+81)=sqrt(106)≈10.30
(b) Reference angle = tan⁻¹(|-9/-5|)=tan⁻¹(9/5)≈tan⁻¹(1.8)≈1.06
Quadrant III → theta ≈ pi + 1.06 ≈ 4.20
(c) tan⁻¹ gives a reference angle; it doesn’t automatically place you in Quadrant III.
An observer on the ground measures the angle of elevation to a drone to be 28 degrees.
The horizontal distance from the observer to the point directly below the drone is 120 ft.
(a) Write a trigonometric equation for the height of the drone.
(b) Find the height, rounded to one decimal place.
(a) tan(28) = h / 120
(b) h ≈ 120 tan(28) ≈ 63.9 ft
A student says: “sin(x) = 1.2 has two solutions on [0, 2pi) because trig equations always have two solutions.”
(a) Is the student correct?
(b) Explain why using a unit-circle reason.
(a) No.
(b) On the unit circle, sine is a y-value, and y can only be between -1 and 1, so sin(x) can’t be 1.2 → 0 solutions.
Solve on [0, 2pi): 5sin(theta) + 1 = -3.
(a) Isolate sin(theta).
(b) Decide if solutions exist.
(c) If yes, find them; if no, explain.
(a) 5sin(theta)=-4 → sin(theta)=-4/5=-0.8
(b) Solutions exist (value is between -1 and 1).
(c) theta1 = sin⁻¹(-0.8) ≈ -0.93 → positive coterminal: 2pi − 0.93 ≈ 5.35
theta2 = pi − (-0.93) = pi + 0.93 ≈ 4.07
So theta ≈ 4.07, 5.35
Solve on [0, 2pi) with exact answers: cos(theta) =
-sqrt(2)/2
theta = 3pi/4 and 5pi/4
Athena starts at the origin, runs 70 units west, then 50 units south, and stops.
(a) Write the endpoint (x,y).
(b) Find tan(theta), sin(theta), cos(theta (exact).
(c) Find theta in degrees to 1 decimal using an inverse trig function.
(a) (-70, -50)
(b) r = sqrt(70^2+50^2)
=sqrt(7400)
=10sqrt(74)
tan(theta)=(-50)/(-70)=5/7
sin(theta)=y/r = -50/(10sqrt(74)) = -5/sqrt(74)
cos(theta)=x/r = -70/(10*sqrt(74)) = -7/sqrt(74)
(c) Reference angle = tan⁻¹(5/7) ≈ tan⁻¹(0.714...) ≈ 35.5 degrees
Quadrant III → theta ≈ 180 + 35.5 = 215.5 degrees
A 20-ft cable is attached to the top of a vertical pole and anchored to the ground, forming a right triangle.
The base of the cable is 12 ft from the pole.
(a) Write an equation to find the angle the cable makes with the ground.
(b) Find the angle, rounded to one decimal degree.
(c) Explain in one sentence why an inverse trig function is required.
(a) cos(theta) = 12 / 20
(b) theta = cos⁻¹(0.6) ≈ 53.1 degrees
(c) An inverse trig function is needed because the unknown is an angle, not a side length.