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Answer
sin x = –3/5, cos x = –4/5, and tan x = 3/4.
Solution
We are given that tan x = 3/4 and that cos x < 0.
Step 1. Identify the quadrant.
• Since tan x = sin x/cos x is positive, sin x and cos x must have the same sign.
• Given that cos x is negative, it follows that sin x must also be negative.
• Therefore, x lies in the third quadrant.
Step 2. Find sin x and cos x.
Imagine a right triangle where the opposite side = 3 and the adjacent side = 4 (ignoring signs initially). Then, the hypotenuse is
hypotenuse = √(3² + 4²) = √(9 + 16) = √25 = 5.
In the standard triangle (first quadrant), one would have:
sin x = 3/5 and cos x = 4/5.
However, since x is in the third quadrant, both sine and cosine must be negative. Therefore,
sin x = –3/5 and cos x = –4/5.
Step 3. Verify:
• tan x = sin x/cos x = (–3/5)/(–4/5) = 3/4, as given.
• cos x < 0, as required.
Thus, the values satisfy the conditions.
Final Answer:
sin x = –3/5, cos x = –4/5, and tan x = 3/4.
Additionally, one can express all solutions for x as:
x = π + arctan(3/4) + πk (for any integer k)
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Simplify this solution Bonus Knowledge
To find \( x \) when \( \tan x = \frac{3}{4} \) and \( \cos x < 0 \), we identify that this places \( x \) in the second quadrant where tangent is negative (but in this case, we have a positive tangent value, so we can derive the reference angle first). The reference angle \( \theta \) can be found using \( \tan^{-1}\left(\frac{3}{4}\right) \). The angle in the second quadrant will then be \( x = \pi - \theta \). You can find \( x \) explicitly by calculating \( \tan^{-1}\left(\frac{3}{4}\right) \) and subtracting it from \( \pi \). Don't forget, since \( \cos x < 0 \), all our calculations in this quadrant are solid!
