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2 years ago

If sin(A+B)=1/2 & sin(A-B)=1/3 find sin(2A)

Mathematics

1 answer:

timofeeve [1]2 years ago

6 0

Answer:

sin(2A) = (2√2 + √3) / 6

Step-by-step explanation:

2A = (A+B) + (A−B)

sin(2A) = sin((A+B) + (A−B))

Angle sum formula:

sin(2A) = sin(A+B) cos(A−B) + sin(A−B) cos(A+B)

sin(2A) = 1/2 cos(A−B) + 1/3 cos(A+B)

Pythagorean identity:

sin(2A) = 1/2 √[1 − sin²(A−B)] + 1/3 √[1 − sin²(A+B)]

sin(2A) = 1/2 √(1 − 1/9) + 1/3 √(1 − 1/4)

sin(2A) = 1/2 √(8/9) + 1/3 √(3/4)

sin(2A) = 1/3 √2 + 1/6 √3

sin(2A) = (2√2 + √3) / 6

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2 years ago

If cos Θ = negative 4 over 7, what are the values of sin Θ and tan Θ? sin Θ = ±square root 33 over 7; tan Θ = ±square root 33 ov

Natali5045456 [20]

Answer:

Answer = Option A;

sinθ = ± $\frac{\sqrt{33} }{7}$

tanθ = ± $\frac{\sqrt{33} }{4}$

Step-by-step explanation:

cosθ = $-\frac{4}{7}$

Remove the negative sign and evaluate cosθ = 4/7

cosθ = adjacent/hypotenuse.

Please see attached image for the right-angle triangle representation.

Let opp = length of opposite side of the triangle.

Let adj = length of adjacent side of the triangle.

Let hyp = length of hypotenuse side of the triangle

Using Pythagoras theorem,

$hyp^{2}= opp^{2} +adj^{2} ;\\\\7^{2}= opp^{2}+ 4^{2};\\ \\opp^{2} =7^{2}- 4^{2} \\\\opp^{2} = 49-16\\\\opp^{2} = 33;$

Take the square root of both sides.

$\sqrt{opp^{2} } =$ ± $\sqrt{33}$

opp = ± $\sqrt{33}$

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tanθ = opposite/adjacent

tanθ = ± $\frac{\sqrt{33} }{4}$

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2 years ago