Question

note: triangle may not be drawn to scale.
suppose a = 2 and b = 8.
find an exact value or give at least two decimal places:
sin(a) =
cos(a) =
tan(a) =
sec(a) =
csc(a) =
cot(a) =
question help: video

Explanation:

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

By Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\). Given \(a = 2\) and \(b=8\), then \(c=\sqrt{2^{2}+8^{2}}=\sqrt{4 + 64}=\sqrt{68}=2\sqrt{17}\)

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

\(\sin(A)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{a}{c}\). Substitute \(a = 2\) and \(c = 2\sqrt{17}\), \(\sin(A)=\frac{2}{2\sqrt{17}}=\frac{1}{\sqrt{17}}\approx0.24\)

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

\(\cos(A)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{b}{c}\). Substitute \(b = 8\) and \(c = 2\sqrt{17}\), \(\cos(A)=\frac{8}{2\sqrt{17}}=\frac{4}{\sqrt{17}}\approx0.97\)

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

\(\tan(A)=\frac{\text{opposite}}{\text{adjacent}}=\frac{a}{b}\). Substitute \(a = 2\) and \(b = 8\), \(\tan(A)=\frac{2}{8}=0.25\)

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

\(\sec(A)=\frac{1}{\cos(A)}\). Since \(\cos(A)=\frac{4}{\sqrt{17}}\), then \(\sec(A)=\frac{\sqrt{17}}{4}\approx1.03\)

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

\(\csc(A)=\frac{1}{\sin(A)}\). Since \(\sin(A)=\frac{1}{\sqrt{17}}\), then \(\csc(A)=\sqrt{17}\approx4.12\)

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

\(\cot(A)=\frac{1}{\tan(A)}\). Since \(\tan(A)=0.25\), then \(\cot(A) = 4\)

Answer:

\(\sin(A)\approx0.24\), \(\cos(A)\approx0.97\), \(\tan(A)=0.25\), \(\sec(A)\approx1.03\), \(\csc(A)\approx4.12\), \(\cot(A)=4\)