Question
If \( \cos \theta \approx 0.3090 \), which of the following represents approximate values of \( \sin \theta \) and \( \tan \theta \), for \( 0^{\circ}<\theta<90^{\circ} \) ? \( \sin \theta \approx 0.9511 ; \tan \theta \approx 0.3249 \) \( \sin \theta \approx 0.9511 ; \tan \theta \approx 3.0780 \) \( \sin \theta \approx 3.2362 ; \tan \theta \approx 0.0955 \) \( \sin \theta \approx 3.2362 ; \tan \theta \approx 10.4731 \)
Ask by Lawson Allan. in the United States
Feb 28,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
sin θ ≈ 0.9511 and tan θ ≈ 3.0780
Solution
We start with the identity sin²θ + cos²θ = 1. Given cos θ ≈ 0.3090, we have:
sin²θ ≈ 1 − (0.3090)² ≈ 1 − 0.0955 ≈ 0.9045.
Taking the positive square root (since 0° < θ < 90° so sin θ > 0):
sin θ ≈ √0.9045 ≈ 0.9511.
Next, tan θ = sin θ / cos θ, so:
tan θ ≈ 0.9511 / 0.3090 ≈ 3.078.
Thus, the correct option is:
sin θ ≈ 0.9511 ; tan θ ≈ 3.0780.
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Simplify this solution Extra Insights
To find the values of \( \sin \theta \) and \( \tan \theta \) given \( \cos \theta \approx 0.3090 \), we can use the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substituting the given value: \[ \sin^2 \theta + (0.3090)^2 = 1 \implies \sin^2 \theta + 0.0956 \approx 1 \implies \sin^2 \theta \approx 0.9044 \implies \sin \theta \approx 0.9511 \] Now, \( \tan \theta = \frac{\sin \theta}{\cos \theta} \approx \frac{0.9511}{0.3090} \approx 3.0780 \). Thus, the correct approximation values are: \( \sin \theta \approx 0.9511 ; \tan \theta \approx 3.0780 \) So, the answer is the second option.
