1.    sinα = 12/13, tanα = 12/5, cosecα = 13/12

we will now solve question 2 together.

Solution to number 2

tan B = 2/3

tan2B = 4/9

tan2B + 1 = sec2B

$\frac{4}{9}&space;+&space;1$ = sec2B

$\frac{4}{9}&space;+&space;\frac{1}{1}$ = sec2B

lcm is 9

$\frac{4&space;+&space;9}{9}&space;=&space;sec^2B&space;\\&space;\frac{13}{9}&space;=&space;sec^2B&space;\\&space;\\&space;\sqrt{\frac{13}{9}}&space;=&space;secB$

root 9 is 3

$\frac{\sqrt{13}}{3}&space;=&space;secB$   but cosB = 1/secB

cosB = $\frac{1}{\sqrt{13}/3}$

$\therefore&space;cosB&space;=&space;\frac{3}{\sqrt{13}}$

since we already know tanB and cosB, we can easily get sinB using

tanB = sinB/cosB

$tanB&space;=&space;\frac{sinB}{cosB}&space;\\&space;\frac{2}{3}&space;=&space;\frac{sinB}{3/\sqrt{13}}&space;\\&space;\frac{2}{3}&space;=&space;sinB&space;\div&space;\frac{3}{\sqrt{13}}\\&space;\frac{2}{3}&space;=&space;\frac{sinB}{1}&space;\div&space;\frac{3}{\sqrt{13}}\\&space;\\&space;\frac{2}{3}&space;=&space;\frac{sinB}{1}&space;\times&space;\frac{\sqrt{13}}{3}$

$\frac{2}{3}&space;=&space;\frac{\sqrt{13}&space;\&space;sinB}{3}$

3 cancel 3 at the denominator

$2&space;=&space;\sqrt{13}&space;\&space;sinB$

divide both sides by $\sqrt{13}$

$\therefore&space;\frac{2}{\sqrt{13}}&space;=&space;sinB$.

EXAMPLE

If cosA = 0.8, find the value of sinA and cotA.

Solution

Sin2A + cos2A = 1

Sin2A + (0.8)2 = 1

Sin2A + 0.64 = 1 (collect like terms)

Sin2A = 1 – 0.64

Sin2A = 0.36 (take the square root of both sides)

$\sqrt{Sin^2A}&space;\&space;=&space;\sqrt{0.36}$

sinA = 0.6

cotA = $\frac{cosA}{sinA}$

cot A = $\frac{0.8}{0.6}$

cot A = 1.33

we can also get our cot a from

1 + cot2A = cosec2A

1 + cot2A = $\frac{1}{sin^2A}$           (cosec2A  = $\frac{1}{sin^2A}$ )

1 + cot2A = $\frac{1}{0.36}$                  (Sin2A = 0.36 before taking the roots of both sides from above)

1 + cot2A = 2.7778

collect like terms

cot2A = 2.7778 - 1

cot2A = 1.7778

take the square root of both sides

$\sqrt{cot^2A}&space;=&space;\sqrt{1.7778}$

cotA = 1.333 (same as our answer above)

Exercise B

Using trigonometric formulas only to solve these problems

1.   If cos A = $\frac{5}{13}$, find tan A and cosec2A

2.   Given that tan B = $\frac{8}{15}$, find the value of cos B and cosec B

3.   What is the value of tan C and cos C given that sin C = $\frac{12}{13}$

4.   Given that sec2P = 1/9, find the value of cosP, sin P and cot2P

5.   If cot x = 3.7, evaluate sin x, cos x and sec x.

6.   What is the value of tan2 Y and sec Y when Sin Y = 2/5

7.   If cot θ = 7, find the value of tan θ , sin θ and sec θ.

8.   Find in terms of k sin θ and cos θ given that tan θ = k

9.   If cosec α = $\frac{1}{\beta&space;}$ find in terms of sec β and cot β.

10.   if sec x = $\sqrt{5}$, what is the value of tan x and cosec x.

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