12 multiple-choice questions, progressively harder.
Triangle ABCABCABC is right at CCC. Use AAA as the acute reference angle. Which side is opposite the selected angle?
Solution
Correct answer: C
The opposite leg lies across from the selected acute angle. For angle AAA, that side is BCBCBC.
CACACA is the adjacent leg, and ABABAB is the hypotenuse.
Triangle ABCABCABC is right at CCC. Use AAA as the acute reference angle. Which side is its adjacent leg?
Correct answer: A
The adjacent leg meets the selected angle and the right angle. Thus CACACA is adjacent to AAA.
Although ABABAB also touches AAA, it is the hypotenuse and is excluded from adjacent leg.
Triangle ABCABCABC is right at CCC. Use AAA as the acute reference angle. Which side is the hypotenuse?
Correct answer: D
The hypotenuse is opposite the right angle at CCC, so it is ABABAB.
It stays the hypotenuse when either acute angle is selected. The leg roles can exchange, but the right angle does not move.
Triangle ABCABCABC is right at CCC. Use AAA as the acute reference angle. Find sinA\sin AsinA from these lengths.
BC=9BC=9BC=9
CA=40CA=40CA=40
AB=41AB=41AB=41
Correct answer: B
Sine uses opposite leg divided by hypotenuse.
The other ratios give cosine, tangent, or the reciprocal of the required sine.
Triangle ABCABCABC is right at CCC. Use BBB as the reference angle and find cosB\cos BcosB.
BC=20BC=20BC=20
CA=21CA=21CA=21
AB=29AB=29AB=29
For angle BBB, the adjacent leg is BCBCBC and the hypotenuse is ABABAB.
The 212121-unit leg is opposite BBB. The hypotenuse belongs in the denominator of cosine.
In a right triangle, acute θ\thetaθ has opposite leg 888 units and adjacent leg 333 units. What is tanθ\tan\thetatanθ?
Tangent divides the opposite leg by the adjacent leg.
A tangent can exceed 111. The reciprocal reverses the legs; the expressions with 73\sqrt{73}73 use the hypotenuse instead.
A right triangle is enlarged by factor 444, with its selected acute angle kept fixed. What happens to the sine of that angle?
Both the opposite leg and the hypotenuse are multiplied by 444.
Their common factor cancels, so sine stays unchanged. The other factors confuse a length or squared length with a ratio.
Two nondegenerate right triangles have the same acute reference angle. Why may their corresponding side ratios be equated?
They have a matching right angle and the given matching acute angle, so AA proves similarity.
Similarity gives proportional corresponding sides. It allows different sizes, so congruence, equal side lengths, and equal perimeters need not follow.
What is the exact value of sin30∘\sin30^\circsin30∘?
A 30∘30^\circ30∘-60∘60^\circ60∘-90∘90^\circ90∘ triangle can have opposite leg 111, adjacent leg 3\sqrt33, and hypotenuse 222 relative to 30∘30^\circ30∘.
The other choices are cosine, tangent, or a reversed sine ratio.
What is the exact value of cos45∘\cos45^\circcos45∘?
An equal-leg right triangle can have legs 111 and hypotenuse 2\sqrt22.
Rationalizing gives 2/2\sqrt2/22/2. The value 111 is tangent at 45∘45^\circ45∘, while 2\sqrt22 reverses the cosine ratio.
What is the exact value of tan60∘\tan60^\circtan60∘?
Relative to 60∘60^\circ60∘, a half-equilateral triangle has opposite leg 3\sqrt33 and adjacent leg 111.
Thus its tangent is 3\sqrt33. Reversing the legs gives 3/3\sqrt3/33/3; the remaining values use the hypotenuse.
Which inequality holds for the cosine of every acute angle θ\thetaθ in a nondegenerate right triangle?
The adjacent leg and hypotenuse are positive, and the leg is strictly shorter than the hypotenuse.
So cosine lies strictly between 000 and 111. Neither endpoint belongs to the acute-angle domain.
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