Question

Please help me figure out this trig identities problem where the short leg = 3 and the long leg =
`√(x^2-9)`. Theta is located between the hypotenuse and the short leg.

(a) Find sec and write this as an equation.

(b) Solve the equation from part (a) for theta.

(c) Solve the equation from part (a) for x.

(d) Express
`√(x^2-9)` in terms of theta, simplifying as much as possible.

Answer 1

Answer: This is a quadratic equation, and we can solve it using factoring, completing the square, or the quadratic formula. Without further information, it is not possible to determine the value of x without more context or additional equations.

Step-by-step explanation: (a) To find sec and write this as an equation, we need to use the given information that the short leg is 3 and the long leg is .

Secant (sec) is defined as the reciprocal of the cosine (cos) function. Since cosine is the ratio of the adjacent side to the hypotenuse, we can use the Pythagorean theorem to find the length of the hypotenuse.

Using the Pythagorean theorem: (short leg)^2 + (long leg)^2 = (hypotenuse)^2

Plugging in the given values: 3^2 + x^2 = (hypotenuse)^2

Simplifying: 9 + x^2 = (hypotenuse)^2

Now, we can express sec as the reciprocal of cos:

sec = 1/cos

Since cosine is the ratio of the adjacent side to the hypotenuse, we can write:

cos = x/hypotenuse

Therefore, sec = 1 / (x/hypotenuse) = hypotenuse/x

Substituting the value of the hypotenuse from the Pythagorean theorem, we have:

sec = (9 + x^2)/x

So, the equation for sec is sec = (9 + x^2)/x.

(b) To solve the equation sec = (9 + x^2)/x for theta, we need to understand that sec is the reciprocal of cos, and cos = adjacent/hypotenuse.

Using the given information, we know that the short leg is the adjacent side and the hypotenuse is (9 + x^2)/x.

Therefore, we can write the equation as:

cos = 3 / ((9 + x^2)/x)

Simplifying further, we can multiply the numerator and denominator by x:

cos = (3x) / (9 + x^2)

Now, we can find theta by taking the inverse cosine (arccos) of both sides:

theta = arccos((3x) / (9 + x^2))

(c) To solve the equation sec = (9 + x^2)/x for x, we can multiply both sides by x:

sec * x = 9 + x^2

Rearranging the equation:

x^2 - sec * x + 9 = 0

Answer 2

a.
`\sf sec(\theta) = (x)/(3)`

b.
`\sf \theta = sec^(-1) \left((x)/(3)\right)`

c.
`\sf x = 3sec(\theta)`

d.
`\sf \theta = sin^(-1)\left((√(x^2-9))/(x)\right)`

Explanation:

Given:

In ∆ with respect to θ.

• Short leg (Adjacent) = 3
• Long leg (Opposite)=
  `√(x^2-9)`.
• Hypotenuse = x

Part (a)

To find sec(θ), we use the following definition:

`\sf sec(\theta) = (1)/(\cos(\theta))`

We know that the hypotenuse is the longest side of a right triangle, so the cosine of theta is equal to the length of the short leg divided (Adjacent) by the length of the hypotenuse.

Therefore, we have:

`\begin{aligned} \sf \cos(\theta) & = ( Adjacent)/(Hypotenuse) \\\\ \textsf{ Substitute the value} \\\\ &= (3)/(x) \end{aligned}`

Again,

Substituting this into the definition of sec, we get:

`\sf sec(\theta) = (1)/((3)/(x))\\\\ = (x)/(3)`

Therefore, the equation for sec(θ) is
`\sf (x)/(3)`.

Part (b)

To solve the equation
`\sf sec(\theta)= (x)/(3)` for θ, we take the inverse secant of both sides. This gives us:

`\sf \theta = sec^(-1) \left((x)/(3)\right)`

However, the inverse secant function is not defined for all values of x/3. For example,
`\sf sec^(-1)(0)` is undefined because the secant function is never equal to 0.

Therefore, the solution to the equation
`\sf sec(\theta)= (x)/(3)` is θ =
`\sf sec^(-1)(0)`, where
`(x)/(3)` is greater than or equal to 1.

Part (c)

To solve the equation
`\sf sec(\theta)= (x)/(3)` for x, we multiply both sides by 3.

Multiplying both sides by 3, we get:

`\sf sec(\theta) * 3= (x)/(3) * 3`

`\sf x = 3sec(\theta)`

Therefore, the solution to the equation
`\sf sec(\theta)= (x)/(3)` for x is x = 3sec(θ).

Part (D)

To express
`\sf √(x^2-9)` in terms of θ , we can use the following equation:

`\sf sin(\theta) = ( opposite )/( hypotenuse)`

We know that the opposite side is
`\sf √(x^2-9)` and the hypotenuse is x, so we can substitute these values into the equation to get:

`\sf sin(\theta) = (√(x^2-9))/(x)`

To solve this equation for θ, we take the inverse sine of both sides. This gives us:

`\sf \theta = sin^(-1)\left((√(x^2-9))/(x)\right)`

Therefore,

`√(x^2-9)` in terms of theta is:

`\sf \theta = sin^(-1)\left((√(x^2-9))/(x)\right)`