Write the given expression in terms of x and y only tan(sin−1 x + cos−1 y)

Answers

Answer 1

Answer:

Answer:

$\tan \left ( \sin ^(-1)x+ \cos ^(-1)y\right )=(xy+√(\left ( 1-x^2 \right )\left ( 1-y^2 \right )))/(y√(1-x^2)-x√(1-y^2))$

Step-by-step explanation:

We need to express $\tan \left ( \sin ^(-1)x+ \cos ^(-1)y\right )$ in terms of x and y .

Let $\sin ^(-1)x=\theta \,,\,\cos ^(-1)y=\phi$ , we get

$x=\sin \theta \,,\,y=\cos \phi$

Formulae Used:

$\sin ^2\theta +\cos ^2\theta =1\n\sin ^2 \phi +\cos ^2 \phi =1$

$\cos \theta =√(1-\sin^2 \theta)=√(1-x^2)\n\sin \phi=√(1-\cos ^2 \phi)=√(1-y^2)$

$\tan \theta =(\sin \theta )/(\cos\theta )=(x)/(√(1-x^2))\n\tan \phi =(\sin \phi)/(\cos\phi)=(√(1-y^2))/(y)$

We know that $\tan \left ( \theta +\phi\right )=(\tan \theta +\tan \phi )/(1-\tan \theta \,\tan \phi )$

$\tan \left ( \sin ^(-1)x+ \cos ^(-1)y\right )=((x)/(√(1-x^2))+(√(1-y^2))/(y))/(1-(x√(1-y^2))/(y√(-x^2)))\n\therefore \tan \left ( \sin ^(-1)x+ \cos ^(-1)y\right )=(xy+√(\left ( 1-x^2 \right )\left ( 1-y^2 \right )))/(y√(1-x^2)-x√(1-y^2))$

Related Questions

The volume of a cubical box is 10.648 cm cube. find the side of the cube.
How do I solve quadratic using the quadratic formula
Find the discount if a $69.99 item is on sale for $39.99
If you save $3,000 at an interest rate of 14 percent per year, how much will you have at the end of six years? A. $5,805.96 B. $6,502.34 C. $6,584.91
the sum of two numbers is 60. the larger number is 12 more than the smaller number. what is the number?

A water taxi travels around an island in a path that can be modeled by the equation y =0.5(x - 16). A water skier is skiing along a path that begins at the point(6,6) and ends at the point (8. - 4).
a. Write a system of equations to model the problem.
b. Is it possible that the water skier could collide with the taxi? Explain

Answers

Final answer:

The system of equations to model the problem is y = 0.5(x - 16) for the water taxi's path and y = -5x + 36 for the water skier's path. No collision is possible between the water skier and the taxi.

Explanation:

a. System of Equations:

To model the problem, we need to find the equation of the water skier's path.

Using the two given points, we can find the slope of the water skier's path:

Find the change in y-coordinates: -4 - 6 = -10
Find the change in x-coordinates: 8 - 6 = 2
Calculate the slope: slope = -10 / 2 = -5

Now that we have the slope, we can use the point-slope formula to find the equation of the water skier's path:

y - y1 = m(x - x1)

Substituting the values x1 = 6, y1 = 6, and m = -5:

y - 6 = -5(x - 6)

Simplifying:

y = -5x + 36

Therefore, the system of equations to model the problem is:

y = 0.5(x - 16) - Equation of the water taxi's path

y = -5x + 36 - Equation of the water skier's path

b. Possibility of Collision:

To determine if the water skier could collide with the taxi, we need to solve the system of equations. If the x-coordinate at the point of intersection is within a certain range, then a collision is possible.

Substitute the second equation into the first equation:

0.5(x - 16) = -5x + 36

Solve for x:

0.5x - 8 = -5x + 36

5.5x = 44

x = 8

The x-coordinate of the point of intersection is 8. Since the water skier's path does not pass through the point x = 8, there is no collision between the water skier and the taxi.

Learn more about System of equations here:

brainly.com/question/35467992

#SPJ2

I realize that the equation is different, but do it in the same way, it'll help! It is given that the water taxi's path can be modeled by the equation y =0.5(x - 14)^2. Therefore, this is one of the equations in this system. Find a linear equation that will model the path of the water skier, which begins at the point (6,6) and ends at the point (8,-4). The slope is (-5). Use the slope and one point on the line to find the y-intercept of the line. The y-intercept of the line that passes through the points (6,6) and (8,-4) is (0,36). Thus, the equation is y=-5x+36. Now, to determine if it is possible for the water skier to collide with the taxi, we have to determine if there is a solution to the system of equations. To determine if there is a solution to the system of equations, solve the system using substitution. First, write the equation that models the water taxi's path in standard form. y=0.5(x - 14)^2-->0.5x^2-14x+98. Use substitution. Substitute for y in the equation and then solve for x. As the expression on the left side of the equation cannot easily be factored, use the Quadratic Formula to solve for x. Do x=-b(plusorminus)sqrrtb^2-4ac/2a. Identify a, b, and c. a=0.5, b=-9, and c=62. Substitute into the Quadratic Formula. If there is a negative number under the radical, there are NO solutions. Thus, the path of the water skier will never cross the path of the taxi.

In conclusion: It is not possible that the water skier could collide with the taxi as the two paths never cross.

Write a word phrase for the algebraic expression 3x-7?

Answers

three x minus seven would be the answer to this question

The legs of a right triangle measure 5 centimeters and 12 centimeters. What is the length of the hypotenuse?

Answers

13cm ... bc , a^2 + b^2 = c^2

a^2 + b^2 = c^2 (pretend the ^2 is a squared symbol)
5^2 + 12^2 = c^2, which is...
25 + 144 = c^2
25 + 144 = 169
So, 169 is c^2 (c squared).
Now, the square root of 169 is 13.
That means the hypotenuse is 13 cma^2 + b^2 = c^2 (pretend the ^2 is a squared symbol)
5^2 + 12^2 = c^2, which is...
25 + 144 = c^2
25 + 144 = 169
So, 169 is c^2 (c squared).
Now, the square root of 169 is 13.
That means the hypotenuse is 13 cm

Answer:

Step-by-step explanation:

a^2 + b^2 = c^2 (pretend the ^2 is a squared symbol)

5^2 + 12^2 = c^2, which is...

25 + 144 = c^2

25 + 144 = 169

So, 169 is c^2 (c squared).

Now, the square root of 169 is 13.

That means the hypotenuse is 13 cma^2 + b^2 = c^2 (pretend the ^2 is a squared symbol)

5^2 + 12^2 = c^2, which is...

25 + 144 = c^2

25 + 144 = 169