Question

find each value to four decimal places. 1. $cos14^{circ}10$ 2. $csc41^{circ}$ 3. $csc43^{circ}30$ 4. $sec25^{circ}50$ using the figure, find angles in degrees and minutes

Explanation:

Step1: Convert minutes to degrees

We know that $1^{\circ}=60'$, so for $14^{\circ}10'$, $10'=\frac{10}{60}\approx0.1667^{\circ}$, then $14^{\circ}10'\approx14.1667^{\circ}$. For $43^{\circ}30'$, $30'=\frac{30}{60} = 0.5^{\circ}$, then $43^{\circ}30'=43.5^{\circ}$. For $25^{\circ}50'$, $50'=\frac{50}{60}\approx0.8333^{\circ}$, then $25^{\circ}50'\approx25.8333^{\circ}$.

Step2: Calculate cosine value

Using a scientific - calculator, $\cos(14.1667^{\circ})\approx0.9703$.

Step3: Calculate cosecant value for $41^{\circ}$

Since $\csc\theta=\frac{1}{\sin\theta}$, $\sin(41^{\circ})\approx0.6561$, then $\csc(41^{\circ})=\frac{1}{\sin(41^{\circ})}\approx\frac{1}{0.6561}\approx1.5242$.

Step4: Calculate cosecant value for $43.5^{\circ}$

$\sin(43.5^{\circ})\approx0.6884$, then $\csc(43.5^{\circ})=\frac{1}{\sin(43.5^{\circ})}\approx\frac{1}{0.6884}\approx1.4526$.

Step5: Calculate secant value for $25.8333^{\circ}$

Since $\sec\theta=\frac{1}{\cos\theta}$, $\cos(25.8333^{\circ})\approx0.9009$, then $\sec(25.8333^{\circ})=\frac{1}{\cos(25.8333^{\circ})}\approx\frac{1}{0.9009}\approx1.1100$.

Answer:

1. $\cos(14^{\circ}10')\approx0.9703$
2. $\csc(41^{\circ})\approx1.5242$
3. $\csc(43^{\circ}30')\approx1.4526$
4. $\sec(25^{\circ}50')\approx1.1100$