Answer:
(e) √(x² -1)
Step-by-step explanation:
You can look up the Pythagorean theorem, and SOH CAH TOA.
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<h3>Pythagorean theorem</h3>
This tells you the relations between the sides of a right triangle. For the purpose of finding tan(θ), you need to know the length of the side that is not marked on the diagram. It is found using the Pythagorean theorem.
The sum of squares of the legs is equal to the square of the hypotenuse. If we call the unknown leg "o", then ...
1² +o² = x² . . . . . the Pythagorean relation
o² = x² -1 . . . . . . subtract 1
o = √(x² -1) . . . . take the square root
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<h3>SOH CAH TOA</h3>
This mnemonic reminds you of the relationship between trig functions and side lengths of a right triangle.
- Sin = Opposite/Hypotenuse
- Cos = Adjacent/Hypotenuse
- Tan = Opposite/Adjacent
This tells you that tan(θ) is the ratio of the side opposite θ to the side adjacent:
tan(θ) = o/1 = o
From the previous section, o = √(x²-1), so ...
tan(θ) = √(x² -1) . . . . . . . matches choice (e)
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<em>Additional comment</em>
Another way to think about this is in terms of the trig identities you have learned:
sec(θ) = 1/cos(θ) = 1/(1/x) = x
sec²(θ) = tan²(θ) +1 ⇒ tan(θ) = √(sec²(θ) -1) = √(x² -1)
