What is the value of \(\sin(90° - A)\)?
Answer: B
This is one of the fundamental complementary angle identities. The sine of an angle is equal to the cosine of its complement.
Therefore, \(\sin(90° - A) = \cos A\).
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If \(\sin \theta = \frac{3}{5}\), what is the value of \(\cos \theta\)?
Answer: B
We use the fundamental trigonometric identity \(\sin^2 \theta + \cos^2 \theta = 1\).
\((\frac{3}{5})^2 + \cos^2 \theta = 1\)
\(\frac{9}{25} + \cos^2 \theta = 1\)
\(\cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}\)
\(\cos \theta = \sqrt{\frac{16}{25}} = \frac{4}{5}\) (Assuming θ is in the first quadrant).
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What is the maximum value of \(3\sin \theta + 4\cos \theta\)?
Answer: C
For an expression of the form \(a\sin \theta + b\cos \theta\), the maximum value is \(\sqrt{a^2+b^2}\) and the minimum value is \(-\sqrt{a^2+b^2}\).
Here, a = 3 and b = 4.
Maximum value = \(\sqrt{3^2+4^2} = \sqrt{9+16} = \sqrt{25} = 5\).
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If \(\sec \theta + \tan \theta = x\), then \(\sec \theta\) is equal to:
Answer: A
We know that \(\sec^2 \theta - \tan^2 \theta = 1\), which can be factored as \((\sec \theta - \tan \theta)(\sec \theta + \tan \theta) = 1\).
Given \(\sec \theta + \tan \theta = x\), we have \(\sec \theta - \tan \theta = \frac{1}{x}\).
Now we have a system of two linear equations:
1) \(\sec \theta + \tan \theta = x\)
2) \(\sec \theta - \tan \theta = 1/x\)
Adding the two equations gives: \(2\sec \theta = x + \frac{1}{x} = \frac{x^2+1}{x}\).
Therefore, \(\sec \theta = \frac{x^2+1}{2x}\).
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The value of \(\frac{1-\tan^2 45°}{1+\tan^2 45°}\) is:
Answer: D
We know that \(\tan 45° = 1\).
Substitute this value into the expression:
\(\frac{1 - (1)^2}{1 + (1)^2} = \frac{1 - 1}{1 + 1} = \frac{0}{2} = 0\).
Alternatively, this is the formula for \(\cos(2\theta)\), so it equals \(\cos(2 \times 45°) = \cos 90° = 0\).
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If \(\sin A = \frac{12}{13}\), find the value of \(\sec A\).
Answer: B
Given \(\sin A = \frac{12}{13}\) (Opposite/Hypotenuse). We can find the adjacent side using the Pythagorean theorem: \(Adj = \sqrt{13^2 - 12^2} = \sqrt{169 - 144} = \sqrt{25} = 5\).
\(\cos A = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{5}{13}\).
\(\sec A = \frac{1}{\cos A} = \frac{13}{5}\).
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If \(\tan \theta = \frac{4}{3}\), then the value of \(\frac{3\sin \theta + 2\cos \theta}{3\sin \theta - 2\cos \theta}\) is:
Answer: B
We can divide both the numerator and the denominator by \(\cos \theta\).
Expression = \(\frac{3(\frac{\sin \theta}{\cos \theta}) + 2(\frac{\cos \theta}{\cos \theta})}{3(\frac{\sin \theta}{\cos \theta}) - 2(\frac{\cos \theta}{\cos \theta})} = \frac{3\tan \theta + 2}{3\tan \theta - 2}\).
Substitute \(\tan \theta = 4/3\):
\(\frac{3(4/3) + 2}{3(4/3) - 2} = \frac{4+2}{4-2} = \frac{6}{2} = 3\).
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The value of \(\frac{\sin 18°}{\cos 72°}\) is:
Answer: B
We use the complementary angle identity \(\cos(90° - \theta) = \sin \theta\).
So, \(\cos 72° = \cos(90° - 18°) = \sin 18°\).
The expression becomes \(\frac{\sin 18°}{\sin 18°} = 1\).
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What is the value of \(\tan 75° + \cot 75°\)?
Answer: C
First, find the values of \(\tan 75°\) and \(\cot 75°\).
\(\tan 75° = \tan(45°+30°) = \frac{\tan 45 + \tan 30}{1 - \tan 45 \tan 30} = \frac{1+1/\sqrt{3}}{1-1/\sqrt{3}} = \frac{\sqrt{3}+1}{\sqrt{3}-1} = 2+\sqrt{3}\).
\(\cot 75° = 1/\tan 75° = \frac{1}{2+\sqrt{3}} = 2-\sqrt{3}\).
The sum is \((2+\sqrt{3}) + (2-\sqrt{3}) = 4\).
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A ramp for a wheelchair has an angle of elevation of 5°. If the ramp is 12 feet long, how high off the ground is the top of the ramp? (Given sin 5° ≈ 0.087)
Answer: B
Let h be the height and L be the length of the ramp.
\(\sin(\text{angle}) = \frac{\text{height}}{\text{length}}\)
\(\sin 5° = \frac{h}{12}\)
h = 12 × \(\sin 5°\) ≈ 12 × 0.087 = 1.044 feet.
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