12 multiple-choice questions, progressively harder.
The cosine of 42∘42^\circ42∘ is approximately 0.7430.7430.743. Which value approximates cos138∘\cos138^\circcos138∘ to the same precision?
Solution
Correct answer: B
The two turns add to 180∘180^\circ180∘. Their points are reflections in the vertical axis, so the horizontal coordinates have opposite signs.
A 170170170 cm arm starts at the origin and follows a ray whose unit point is (−1385,8485)\left(-\tfrac{13}{85},\tfrac{84}{85}\right)(−8513,8584). Where is its tip?
Correct answer: D
The tip lies 170170170 times as far along the ray as the unit point. Multiply both coordinates by 170170170.
Two braces of lengths 101010 m and 171717 m enclose an angle of 135∘135^\circ135∘. Find the exact area of their triangle.
Correct answer: A
Use the sine of the acute supplement, 45∘45^\circ45∘.
In triangle ABCABCABC, point AAA is the origin and BBB is 666 units to the right of AAA. Point CCC has coordinates (11,7)(11,7)(11,7). Where is the foot of the perpendicular from CCC to line ABABAB?
Correct answer: C
The perpendicular to the horizontal base line is vertical. It lands at (11,0)(11,0)(11,0), which is to the right of BBB at (6,0)(6,0)(6,0).
Angle CCC is the angle measured in triangle ABCABCABC. Which two opposite-side labels give the lengths that can be used with sinC\sin CsinC in the area formula?
The sides that meet at CCC are BCBCBC and ACACAC. Their opposite vertices are AAA and BBB, so their lengths are aaa and bbb.
Two straight edges of a triangular stage are 191919 m and 313131 m long. They meet at 108∘108^\circ108∘. What is the area to the nearest tenth of a square meter?
Use the included angle in degree mode. Round after the multiplication.
A triangle has area 90 cm290\text{ cm}^290 cm2. Two sides meet at 30∘30^\circ30∘, and one of those sides is 242424 cm long. How long is the other?
Call the unknown length uuu. Use the area formula and the exact sine of 30∘30^\circ30∘.
In triangle ABCABCABC, side ABABAB has length 111111 and side ACACAC has length 181818. The base extends left from AAA through EEE, and ∠EAC\angle EAC∠EAC is 64∘64^\circ64∘. Which expression gives the area?
The interior angle at AAA is supplementary to ∠EAC\angle EAC∠EAC. The two angles have the same sine. Half the product of the given sides is 999999.
Two triangles have horizontal bases of lengths 101010 and 252525. In each, a 141414-unit side rises from the left end of its base. The two included angles are 51∘51^\circ51∘ and 129∘129^\circ129∘, respectively. What is the ratio of the first area to the second?
The sides of length 141414 make supplementary turns, so they reach equal heights. The common height and the factor of one half cancel in the area ratio.
A segment of length 737373 makes an acute angle θ\thetaθ from the positive horizontal axis. Its endpoint is 484848 units above that axis. What is its signed horizontal projection?
The vertical leg has length 484848 and the hypotenuse has length 737373. The projection xxx is positive because the angle is acute.
The same two side lengths form two different triangles, with included angles 32∘32^\circ32∘ and 103∘103^\circ103∘. Which triangle has greater area?
The sine at 103∘103^\circ103∘ is the same as at 77∘77^\circ77∘. From 32∘32^\circ32∘ to 77∘77^\circ77∘, the unit point rises toward the top of the semicircle. The 103∘103^\circ103∘ triangle therefore has the greater height and area for the same side lengths.
Two sides of lengths 666 cm and 131313 cm rotate about their common endpoint. Which proposed area is impossible for their triangle?
The greatest height is the full length of the rotating side at 90∘90^\circ90∘. The maximum area is 39 cm239\text{ cm}^239 cm2. As the hinge opens from 0∘0^\circ0∘ to 90∘90^\circ90∘, the height passes through every value from zero to that maximum height.
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