Unit 11: Trigonometric Functions
Angles wrapped around a circle of radius one, and the wave that comes out when you unroll that circle into a line.
8 topics · 35 questions
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Angles in Standard Position and Radian Measure
- How it worksHow does an angle's arc length on a unit circle define its measure in radians?
- Worked exampleFind the radian measure of 210 degrees and the degree measure of 5 π over 6 radians.
- Common mistakeWhy isn't a radian just an arbitrary unit the way a degree is?
Your turn1 of 2
Find the radian measure of 239 degrees.The Unit Circle Definition of Sine and Cosine
- How it worksHow do the x and y coordinates of a point on the unit circle become cosine and sine of its angle?
- Worked exampleFind sine and cosine of 2 π over 3 using the unit circle.
- Common mistakeWhy isn't sine of an angle just defined as opposite over hypotenuse once the angle goes past 90 degrees?
- Going furtherWhat happens to a point's coordinates on the unit circle once the angle passes 2 π or goes negative?
- Ties togetherHow does the unit circle definition of sine and cosine match the right-triangle definition once the angle is acute?
Your turn1 of 2
Using the unit circle, find cos(4.24) + sin(4.24), where 4.24 is given in radians.Special Angles on the Unit Circle
- How it worksHow do the 30-60-90 and 45-45-90 triangles produce the exact sine and cosine values at every special angle?
- Worked exampleFind the exact values of sine and cosine at π over 4 and at π over 3.
- Common mistakeWhy doesn't the sine of 30 degrees just equal half of the sine of 60 degrees?
- Going furtherWhat happens to a 30-60-90 triangle's special angle values once you reflect the angle into the second quadrant?
- Ties togetherHow do these special angle values let you find exact sine and cosine at an angle like 2 π over 3 without a calculator?
Your turn1 of 2
Find the value of 7 times sin(π/3) plus 6 times cos(π/4).The Unit Circle Unrolling into a Sine Wave
- How it worksWatch the unit circle unroll along a straight line while its height at every angle traces the sine graph.
- Worked exampleTrace the height of the point at angle theta on the unit circle for theta from 0 to 2 π and match it to the sine graph.
- Common mistakeWhy doesn't unrolling the unit circle produce a straight line, given that the point moves at a constant angular speed?
- Going furtherWhat graph do you get if you unroll the unit circle by tracking the x-coordinate instead of the height?
- Ties togetherHow does the unit circle's period of 2 π connect to the sine wave's period on its graph?
Your turn1 of 2
A point on the unit circle is at angle x radians. Write its height (y-coordinate) above the center, as a function of x.Graphing Sine and Cosine Functions
- How it worksHow do amplitude, period, phase shift, and vertical shift reshape the basic sine wave?
- Worked exampleGraph y = 3 sin(2x - π over 2) + 1, identifying its amplitude, period, phase shift, and midline.
- Common mistakeWhy doesn't increasing the amplitude of a sine function change how often it repeats?
- Ties togetherHow do the amplitude and period here connect to the same a, h, and k transformation parameters from other parent functions?
Your turn1 of 2
Find the maximum value of y = 7 sin(3x - 4) + 3.The Tangent Function and Its Graph
- How it worksHow does tangent being sine divided by cosine create a vertical asymptote everywhere cosine hits zero?
- Worked exampleFind every vertical asymptote of y = tan(x) between 0 and 2 π.
- Common mistakeWhy doesn't the tangent function have the same period as sine and cosine?
- Going furtherWhat does the tangent function's graph do between two consecutive vertical asymptotes?
Your turn1 of 2
Simplify tan(x) times cos(x), as a function of x.Solving Basic Trigonometric Equations
- How it worksHow does the unit circle reveal every angle that solves an equation like sin(x) = one half?
- Worked exampleSolve sin(x) = one half for x on the interval from 0 to 2 π.
- Common mistakeWhy isn't the reference angle the only solution to a trig equation on a full circle?
- Going furtherWhat happens to the set of solutions when the same trig equation is solved over all real numbers instead of one circle?
Your turn1 of 2
Solve sin(x) = 0.618 for x on the interval from 0 to π/2 (give the value of x in radians).The Law of Sines and the Law of Cosines
- How it worksHow does the law of cosines generalize the Pythagorean theorem to triangles without a right angle?
- Worked exampleFind the third side of a triangle with sides 7 and 10 and an included angle of 50 degrees using the law of cosines.
- Worked exampleFind the two possible measures of angle B in a triangle with side a = 8, side b = 11, and angle A = 40 degrees using the law of sines.
- Common mistakeWhy doesn't the law of sines work directly when you only know two sides and the angle between them?
- Going furtherWhat happens when the law of sines gives two different possible triangles for the same given measurements?
Your turn1 of 2
Find the third side of a triangle with sides 11 and 11 and an included angle of 151 degrees, using the law of cosines.