Question
If \( \tan \theta=\frac{3}{4} \), fnd \( \cos \theta \)
Ask by Chambers Perkins. in Nigeria
Feb 17,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( \cos \theta = \frac{4}{5} \)
Solution
We start with the given:
tan θ = 3/4
Recall that tan θ is defined as sin θ divided by cos θ. That is:
tan θ = sin θ / cos θ
One way to handle this is to imagine a right triangle where the side opposite the angle θ is 3 and the side adjacent is 4. By the Pythagorean theorem, the hypotenuse (h) is:
h = √(3² + 4²) = √(9 + 16) = √25 = 5
Since cosine is the ratio of the adjacent side to the hypotenuse, we get:
cos θ = 4/5
Note that unless otherwise specified, we assume θ is an acute angle, in which case cos θ is positive.
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Simplify this solution Bonus Knowledge
To find \( \cos \theta \) when \( \tan \theta = \frac{3}{4} \), we can use the identity \( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \). Here, we can think of a right triangle where the opposite side is 3 and the adjacent side is 4. Using the Pythagorean theorem, we can find the hypotenuse \( h \): \[ h = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. \] Now, we can find \( \cos \theta \) using \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \): \[ \cos \theta = \frac{4}{5}. \] So, \( \cos \theta = \frac{4}{5} \).
