Answer: 

           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} + 1 = sec2B

\frac{4}{9} + \frac{1}{1} = sec2B

lcm is 9

\frac{4 + 9}{9} = sec^2B \\ \frac{13}{9} = sec^2B \\ \\ \sqrt{\frac{13}{9}} = secB

root 9 is 3

\frac{\sqrt{13}}{3} = secB   but cosB = 1/secB

cosB = \frac{1}{\sqrt{13}/3}

\therefore cosB = \frac{3}{\sqrt{13}}

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

tanB = sinB/cosB

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

\frac{2}{3} = \frac{\sqrt{13} \ sinB}{3}

3 cancel 3 at the denominator

2 = \sqrt{13} \ sinB

divide both sides by \sqrt{13}

\therefore \frac{2}{\sqrt{13}} = 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} \ = \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} = \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 } 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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