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katrin [286]
3 years ago
5

Find the exact value of sin(u+v) given that sin u=7/25 and cos v=-12/13

Mathematics
2 answers:
Yuki888 [10]3 years ago
7 0

Answer:

Step-by-step explanation:

The sum identity for sin(u + v) is:

sin(u + v) = sin(u)cos(v) + cos(u)sin(v)

We were given the value for sin(u) and for cos(v).  But if you notice in the identity, we need sin(v) and cos(u).  We will find those and then fit them into the identity and solve.  First for the cos(u):

The sin ratio is the side opposite the reference angle over the hypotenuse in a right triangle.  If we set the reference angle in standard position (at the origin), and we are in QI, then the side opposite the reference angle is 7 and the hypotenuse is 25, so we need to find the side adjacent to the reference angle.  We will do that using Pythagorean's Theorem:

7^2+x^2=25^2 and

x^2=25^2-7^2 and

x^2=625-49 and

x^2=576 so taking the square root of both sides gives us that

x = 24.

If x = 24 and that is the side adjacent to the reference angle, then

cos(u)=\frac{24}{25}

We will do the same to find sin(v).  This time we have to be in a quadrant where x is negative.  I set the reference angle in QII since x is negative there.  The reference angle is sitting at the origin, the side adjacent to the reference angle is -12, the hypotenuse is 13, so to find y (the side opposite the reference angle, we will again use Pythagorean's Theorem:

(-12)^2+y^2=13^2 and

y^2=13^2-(-12)^2 and

y^2=169-144 and

y^2=25 so

y = 5 and that is the side across from the reference angle.  That means that

sin(v)=\frac{5}{13}

Now we have all the identities we need:

sin(u)=\frac{7}{25} , cos(u)=\frac{24}{25} , sin(v)=\frac{5}{13} , cos(v)=-\frac{12}{13}

Filling in the formula:

(\frac{7}{25}*-\frac{12}{13})+(\frac{24}{25}*\frac{5}{13}) which simplifies to

-\frac{84}{325}+\frac{120}{325}=\frac{36}{325}

So there you go!

Helen [10]3 years ago
4 0

Answer

Step-by-step explanation:

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the function t(n) 8n represents the number of tires t(n) that are needed for n trucks. if t(n)=200. what is the value of n?
mr_godi [17]

Step-by-step explanation:

if I understand correctly, then the function is

t(x) = 8x

now we know, that for a specific n

t(n) = 200

that means

200 = 8n

n = 200/8 = 25

the function tells us, that for 25 trucks we need 200 tires.

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2 years ago
Is (3,4) a solution to the system of<br> equations. <br> X + y = 7<br> x - 2y = - 5
Anuta_ua [19.1K]

Answer: Yes, the point (3,4) is a solution to the system.

===================================================

Proof of this:

Replace x with 3 and y with 4 in the first equation

x+y = 7

3+4 = 7

7 = 7

This confirms the first equation. Repeat for the second equation

x-2y = -5

3-2(4) = -5

3 - 8 = -5

-5 = -5

We get true equations for both when we plug in (x,y) = (3,4). This confirms it is a valid solution to the system of equations. It turns out it's the only solution to this system of equations. Visually, the two lines cross at the single location (3,4).

8 0
3 years ago
The height of a right triangle is √6 units and the base is √9 units. Find the area of the triangle
In-s [12.5K]

Answer:

Step-by-step explanation:

h = √6 units

b = √9 units

A = b*h/2 = √6·√9 /2 = √(6·9)/2 = √(2·3·3·3)/2 = 3√3 /2 units ²≈ 2.6 units²

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3 years ago
a ship leave port p and sails on a bearing N50 degree E to Port Q 15km away.then it sails on a bearing of S45 degree E to Port R
Shtirlitz [24]

Answer:

Port r is 100° from Port p and 26km from Port p

Step-by-step explanation:

Lets note the dimension.

From p to q = 15 km = a

From q to r = 20 km= b

Angle at q = 50° + 45°

Angle at q = 95°

Ley the unknown distance be x

Distance from p to r is the unknown.

The formula to be applied is

X²= a²+ b² - 2abcosx

X²= 15² + 20² - 2(15)(20)cos95

X²= 225+400-(-52.29)

X²= 677.29

X= 26.02

X is approximately 26 km

To know it's direction from p

20/sin p = 26/sin 95

Sin p= 20/26 * sin 95

Sin p = 0.7663

P= 50°

So port r is (50+50)° from Port p

And 26 km far from p

3 0
3 years ago
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