Question

question 14
note: triangle may not be drawn to scale.
suppose a = 8 and b = 4.
find an exact value or give at least two decimal places:
sin(a) =
cos(a) =
tan(a) =
sec(a) =
csc(a) =
cot(a) =

Explanation:

Step1: Find hypotenuse c

Using Pythagorean theorem $c=\sqrt{a^{2}+b^{2}}=\sqrt{8^{2}+4^{2}}=\sqrt{64 + 16}=\sqrt{80}=4\sqrt{5}$

Step2: Calculate sin(A)

$\sin(A)=\frac{a}{c}=\frac{8}{4\sqrt{5}}=\frac{2\sqrt{5}}{5}\approx0.89$

Step3: Calculate cos(A)

$\cos(A)=\frac{b}{c}=\frac{4}{4\sqrt{5}}=\frac{\sqrt{5}}{5}\approx0.45$

Step4: Calculate tan(A)

$\tan(A)=\frac{a}{b}=\frac{8}{4} = 2$

Step5: Calculate sec(A)

$\sec(A)=\frac{c}{b}=\frac{4\sqrt{5}}{4}=\sqrt{5}\approx2.24$

Step6: Calculate csc(A)

$\csc(A)=\frac{c}{a}=\frac{4\sqrt{5}}{8}=\frac{\sqrt{5}}{2}\approx1.12$

Step7: Calculate cot(A)

$\cot(A)=\frac{b}{a}=\frac{4}{8}=\frac{1}{2}=0.5$

Answer:

$\sin(A)=\frac{2\sqrt{5}}{5}\approx0.89$
$\cos(A)=\frac{\sqrt{5}}{5}\approx0.45$
$\tan(A)=2$
$\sec(A)=\sqrt{5}\approx2.24$
$\csc(A)=\frac{\sqrt{5}}{2}\approx1.12$
$\cot(A)=0.5$