Unit 4: Angles, Radians, and Right Triangle Trigonometry
Measuring rotation in degrees and radians, wrapping a circle's own radius along its arc, and the unit circle that grounds every trig function.
9 topics · 33 questions
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Angles in Standard Position
- How it worksHow do you sketch an angle of 210 degrees in standard position on the coordinate plane?
- Worked exampleFind a coterminal angle between 0 and 360 degrees for an angle of 745 degrees.
- Common mistakeWhy can an angle be negative even though a distance never can be?
Your turn1 of 2
Find the coterminal angle between 0 and 360 degrees for an angle of 607 degrees.Radian Measure
- How it worksHow does one radian come from laying the radius length along the arc of a circle?
- Worked exampleFind the radian measure equivalent to an angle of 150 degrees.
- Common mistakeWhy isn't a radian an arbitrary unit like a degree, since it comes from the circle's own radius?
- Ties togetherHow does defining a radian from arc length make the arc length formula s = r times theta almost obvious?
Your turn1 of 2
Find the radian measure equivalent to an angle of 247 degrees.Converting Between Degrees and Radians
- How it worksHow do you convert between degrees and radians using the ratio π radians over 180 degrees?
- Worked exampleFind the degree measure of an angle of 5pi/6 radians.
- Worked exampleFind the radian measure of an angle of -270 degrees.
- Common mistakeWhy doesn't forgetting to convert to radians silently break a graph's period on a graphing calculator?
Your turn1 of 2
Find the degree measure of an angle of 8pi/8 radians.Arc Length
- How it worksHow does the formula s = r·theta turn a central angle in radians into an arc length?
- Worked exampleFind the arc length cut off by a central angle of 2pi/3 radians on a circle of radius 9 cm.
- Worked exampleFind the arc length cut off by a central angle of 45 degrees on a circle of radius 6 cm.
- Common mistakeWhy won't the arc length formula s = r·theta work if theta is measured in degrees?
Your turn1 of 2
Write the arc length as a function of the central angle theta (in radians) for a circle of radius 11 cm.Linear and Angular Speed
- How it worksHow does linear speed v = r·omega connect a point's distance from the center to how fast it travels?
- Worked exampleFind the linear speed of a point on the rim of a wheel of radius 0.5 m spinning at 3 radians per second.
- Common mistakeWhy doesn't angular speed change for two points on the same spinning disk even when their linear speeds differ?
Your turn1 of 2
Write the linear speed as a function of the angular speed omega for a point on the rim of a wheel of radius 1.42 m.Right Triangle Trigonometry
- How it worksHow do SOH-CAH-TOA connect an acute angle to ratios of a right triangle's sides?
- Worked exampleFind the height of a tree that casts a 40-foot shadow when the sun's angle of elevation is 35 degrees.
- Common mistakeWhy do the trig ratios only apply directly to a right triangle, not to any triangle you draw?
- Common mistakeWhy doesn't SOH-CAH-TOA work directly on an angle that isn't between 0 and 90 degrees?
Your turn1 of 2
Write the height of a tree as a function of the sun's angle of elevation theta (in radians), given a shadow of length 73 feet.The Unit Circle
- How it worksHow do the coordinates of a point on the unit circle become the cosine and sine of its angle?
- Worked exampleFind the coordinates of the point on the unit circle at an angle of 5pi/4.
- Common mistakeWhy isn't the unit circle definition of sine and cosine limited to acute angles the way SOH-CAH-TOA is?
- Ties togetherHow do the coordinates on the unit circle extend from sine and cosine into all six trig functions?
Your turn1 of 2
Find the x-coordinate of the point on the unit circle at an angle of 5pi/5.The Six Trigonometric Functions on the Unit Circle
- How it worksHow do tangent, secant, cosecant, and cotangent build out of ratios of cosine and sine?
- Worked exampleFind all six trig function values for an angle of π/3.
- Common mistakeWhy is tangent undefined at angles where cosine equals zero, not where sine does?
Your turn1 of 2
Find sec(π/7).Reference Angles and Exact Values
- How it worksHow does a reference angle let you find an exact trig value in any quadrant from a first-quadrant angle?
- Worked exampleFind the exact value of cos(210 degrees) using its reference angle.
- Worked exampleFind the exact value of sin(5pi/3) using its reference angle.
- Common mistakeWhy does the sign of a trig value depend on the quadrant even though the reference angle stays the same?
Your turn1 of 2
Find the exact value of cos(211 degrees) using its reference angle.