if $cos\thetaapprox0.3090$, which of the following represents approximate values of $sin\theta$ and…

if $cos\thetaapprox0.3090$, which of the following represents approximate values of $sin\theta$ and $\tan\theta$, for $0^{circ}<\theta<90^{circ}$?\n$sin\thetaapprox0.9511;\tan\thetaapprox0.3249$\n$sin\thetaapprox0.9511;\tan\thetaapprox3.0780$\n$sin\thetaapprox3.2362;\tan\thetaapprox0.0955$\n$sin\thetaapprox3.2362;\tan\thetaapprox10.4731$

if $cos\thetaapprox0.3090$, which of the following represents approximate values of $sin\theta$ and $\tan\theta$, for $0^{circ}<\theta<90^{circ}$?\n$sin\thetaapprox0.9511;\tan\thetaapprox0.3249$\n$sin\thetaapprox0.9511;\tan\thetaapprox3.0780$\n$sin\thetaapprox3.2362;\tan\thetaapprox0.0955$\n$sin\thetaapprox3.2362;\tan\thetaapprox10.4731$

Answer

Explanation:

Step1: Use the Pythagorean identity

We know that $\sin^{2}\theta+\cos^{2}\theta = 1$. Given $\cos\theta\approx0.3090$, then $\sin\theta=\sqrt{1 - \cos^{2}\theta}$. $\sin\theta=\sqrt{1-(0.3090)^{2}}=\sqrt{1 - 0.095481}=\sqrt{0.904519}\approx0.9511$.

Step2: Use the tangent - sine and cosine relationship

The tangent function is defined as $\tan\theta=\frac{\sin\theta}{\cos\theta}$. Since $\sin\theta\approx0.9511$ and $\cos\theta\approx0.3090$, then $\tan\theta=\frac{0.9511}{0.3090}\approx3.0780$.

Answer:

$\sin\theta\approx0.9511;\tan\theta\approx3.0780$