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If sin(x) = 1/3 and sec(y) = 25/24
, where x and y lie between 0 and π/2, evaluate the expression using trigonometric identities. (Enter an exact answer.)

What I know:
cos (y) = 25/24
sin (x + y) = (sinx)(cosy) + (cosx)(siny)
so...
sin (x + y) = (1/3)(24/25) + (cosx)(siny)

How do you get cosx and siny?

Thanks.

Respuesta :

scme1702
We can approach this in another way.
We know that sin(∅) = height / hypotenuse.

Thus, for x, height is 1 and hypotenuse is 3. Using Pythagoras theorem,
3² = 1² + b²
b = √8
cos(x) = b/hypotenuse
cos(x) = √8 / 3

Now, lets consider y:
sec(y) = 1 / cos(y) = 1 / base / hypotenuse = hypotenuse / base
The hypotenuse is 25 and the base is 24. We again apply Pythagoras theorem to find the third side, which works out to be:
height = 7
sin(y) = height / hypotenuse
sin(y) = 7/25

Now, sin(x + y) =
sin(x)cos(y) + sin(y)cos(x)
= (1/3)(24/25) + (√8 / 3)(7/25)
= 8/25 + 7√8/75
= (24 + 14√2) / 75
johanrusli

sin (x + y) ≈ 0.58

Further explanation

Firstly , let us learn about trigonometry in mathematics.

Suppose the ΔABC is a right triangle and ∠A is 90°.

sin ∠A = opposite / hypotenuse

cos ∠A = adjacent / hypotenuse

tan ∠A = opposite / adjacent

There are several trigonometric identities that need to be recalled, i.e.

[tex]cosec ~ A = \frac{1}{sin ~ A}[/tex]

[tex]sec ~ A = \frac{1}{cos ~ A}[/tex]

[tex]cot ~ A = \frac{1}{tan ~ A}[/tex]

[tex]tan ~ A = \frac{sin ~ A}{cos ~ A}[/tex]

Let us now tackle the problem!

If sec (y) = 25/24 , then we can assume that :

adjacent side = 24 cm

hypotenuse = 25 cm

opposite side = [tex]\sqrt{25^2 - 24^2}[/tex] = [tex]7 ~ cm[/tex]

sin (y) = opposite / hypotenuse = [tex]7/25[/tex]

If sin (x) = 1/3 , then we can assume that :

opposite side = 1 cm

hypotenuse = 3 cm

adjacent side = [tex]\sqrt{3^2 - 1^2}[/tex] = [tex]\sqrt{8} ~ cm[/tex]

cos (x) = adjacent / hypotenuse = [tex]\sqrt{8}/3[/tex] = [tex]\frac{2\sqrt{2}}{3}[/tex]

[tex]\sin (x + y) = \sin x ~ \cos y + \cos x ~ \sin y[/tex]

[tex]\sin (x + y) = \frac{1}{3} \times \frac{24}{25} + \frac{2\sqrt{2}}{3} \times \frac{7}{25}[/tex]

[tex]\large {\boxed {\sin (x + y) = \frac{24 + 14\sqrt{2}}{75} \approx 0.58} }[/tex]

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Answer details

Grade: College

Subject: Mathematics

Chapter: Trigonometry

Keywords: Sine , Cosine , Tangent , Opposite , Adjacent , Hypotenuse , Triangle , Fraction , Lowest , Function , Angle

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