Question

standard 14 homework
score: 2/30 answered: 2/18
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question 3
suppose ( a = 4 ) and ( b = 10 ) in the right triangle shown above. evaluate each of the following trigonometric
expressions. give exact answers in fraction form; no decimals allowed.
( sin ( a ) = )
( cos ( a ) = )
( \tan ( a ) = )
( sec ( a ) = )
( csc ( a ) = )
( cot ( a ) = )
question help: video message instructor post to forum

Explanation:

Step1: Find the hypotenuse \(c\) using Pythagorean theorem

$$c=\sqrt{a^{2}+b^{2}}=\sqrt{4^{2}+10^{2}}=\sqrt{16 + 100}=\sqrt{116}=2\sqrt{29}$$

Step2: Calculate \(\sin(A)\)

$$\sin(A)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{a}{c}=\frac{4}{2\sqrt{29}}=\frac{2}{\sqrt{29}}=\frac{2\sqrt{29}}{29}$$

Step3: Calculate \(\cos(A)\)

$$\cos(A)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{b}{c}=\frac{10}{2\sqrt{29}}=\frac{5}{\sqrt{29}}=\frac{5\sqrt{29}}{29}$$

Step4: Calculate \(\tan(A)\)

$$\tan(A)=\frac{\text{opposite}}{\text{adjacent}}=\frac{a}{b}=\frac{4}{10}=\frac{2}{5}$$

Step5: Calculate \(\sec(A)\)

$$\sec(A)=\frac{1}{\cos(A)}=\frac{c}{b}=\frac{2\sqrt{29}}{10}=\frac{\sqrt{29}}{5}$$

Step6: Calculate \(\csc(A)\)

$$\csc(A)=\frac{1}{\sin(A)}=\frac{c}{a}=\frac{2\sqrt{29}}{4}=\frac{\sqrt{29}}{2}$$

Step7: Calculate \(\cot(A)\)

$$\cot(A)=\frac{1}{\tan(A)}=\frac{b}{a}=\frac{10}{4}=\frac{5}{2}$$

Answer:

\(\sin(A)=\frac{2\sqrt{29}}{29}\), \(\cos(A)=\frac{5\sqrt{29}}{29}\), \(\tan(A)=\frac{2}{5}\), \(\sec(A)=\frac{\sqrt{29}}{5}\), \(\csc(A)=\frac{\sqrt{29}}{2}\), \(\cot(A)=\frac{5}{2}\)