12 multiple-choice questions, progressively harder.
Where is the unit point for an angle of 90∘90^\circ90∘?
Solution
Correct answer: C
A quarter-turn from the positive horizontal axis reaches the top of the unit semicircle. The point is one unit above the origin.
The unit point for θ\thetaθ is (20101,99101)\left(\tfrac{20}{101},\tfrac{99}{101}\right)(10120,10199). What is sinθ\sin\thetasinθ?
Correct answer: A
Sine reads the vertical coordinate on the unit semicircle. The second coordinate is 99101\tfrac{99}{101}10199.
For a ray at 161∘161^\circ161∘, which signs do its cosine and sine have, in that order?
Correct answer: D
The ray meets the upper semicircle above the horizontal axis and to the left of the vertical axis. The cosine reads a negative horizontal coordinate. The sine reads a positive vertical coordinate.
Which angle is supplementary to 66∘66^\circ66∘?
Correct answer: B
The two measures add to a straight angle.
Which expression equals sin93∘\sin93^\circsin93∘?
The two angles 93∘93^\circ93∘ and 87∘87^\circ87∘ add to 180∘180^\circ180∘. Their semicircle points have equal heights.
Which expression equals cos112∘\cos112^\circcos112∘?
The supplement of 112∘112^\circ112∘ is 68∘68^\circ68∘. Reflection in the vertical axis changes the sign of the horizontal coordinate.
An arm starts at the origin and ends at (−14,9)(-14,9)(−14,9) in centimeter coordinates. What is its signed horizontal projection?
The projection reads the horizontal coordinate with its sign. The tip is 141414 cm to the left of the origin.
You know the lengths of ACACAC and BCBCBC. Which angle is included between them?
The given sides end at the same vertex, CCC. The two rays from CCC to the other endpoints form the included angle.
Two sides of lengths 777 cm and 161616 cm meet at a right angle. What is the area?
At a right angle, the sides themselves give the base and the height.
In the diagram, the base is 141414 units, the slanted side is 999 units, and the included angle is 35∘35^\circ35∘. Which expression gives the perpendicular height?
Drop the perpendicular from the free end of the slanted side. The 999-unit side is the hypotenuse of the right triangle and the height is opposite 35∘35^\circ35∘.
A triangle has two sides of lengths 888 m and 151515 m. Their included angle measures 30∘30^\circ30∘. What is its area?
Use the sine of the angle where the given sides meet. The sine of 30∘30^\circ30∘ is one half.
The angle between two fixed sides changes from 47∘47^\circ47∘ to 133∘133^\circ133∘. How does the area change?
The two turns are supplementary and have the same sine. Both side lengths also stay fixed, so the area is unchanged.
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