# Finding the equation for a tangent line (Parametric Equation)

• PsychonautQQ
In summary, the conversation discusses finding an equation of the tangent to a curve given by x=tan(θ) and y=sec(θ), at the point (1, sqrt(2)). The steps to finding the equation involve finding the slope of the tangent at the given point, which is determined to be m=sqrt(2)/2. Using this slope and the given point, the final equation of the tangent line is found to be y=(sqrt(2)/2)x+sqrt(2). However, it is important to use the exact form of the equation, rather than a decimal approximation, to ensure the correct answer.

## Homework Statement

Find an equation of the tangent to the curve given by
x=tan(∅)
y=sec(∅)
at the point (1,sqrt(2))

The answer should be in the form y=f(x) without ∅

## The Attempt at a Solution

Tangent line equation...
y-y*=m(x-x*)
m = dy/dx
m = sec(∅)tan(∅) / (sec^2(∅)
m = tan∅ / sec(∅)
m = sin∅

This is what I did so far, I'm a bit lost obviously

Hint: use "θ" for theta, not "∅".

Find out what θ is when x = 1 and y = sqrt(2) (this is from the given point). Plug the value for θ in your expression for dy/dx to get your slope.

Okay, so i did that and end up with m = sqrt(2)/2.
If i use the equation
y-y*=m(x-x*)
with
y* = sqrt(2)
x* = 1 and solve it i get
y=(sqrt(2)/2)x+2.1
but i type this into my online course and it says it's wrong?

PsychonautQQ said:
y=(sqrt(2)/2)x+2.1
I'm not getting 2.1.

When i type in
y=.71x+.71 it gives me the wrong answer as well. ;-(

Maybe it is because you are using a decimal approximation? I'm pretty sure the exact form of the last equation you wrote is correct.

## 1. What is a tangent line?

A tangent line is a line that touches a curve at exactly one point. It represents the slope of the curve at that specific point.

## 2. How do you find the equation for a tangent line?

To find the equation for a tangent line, you need to first determine the slope of the tangent line at the given point. This can be done by finding the derivative of the function at that point, or by using the slope formula. Once you have the slope, you can use the point-slope form or the slope-intercept form to write the equation of the tangent line.

## 3. What is a parametric equation?

A parametric equation is a set of equations that express the coordinates of a point as functions of one or more independent variables, known as parameters. It is often used to represent curves or surfaces in mathematics and physics.

## 4. How do you find the parametric equations of a curve?

To find the parametric equations of a curve, you need to first determine the coordinates of points on the curve. These points can be found by plugging in different values for the parameter(s) in the original equation. Once you have a set of points, you can use them to create a table and find a pattern. This pattern can then be used to write the parametric equations.

## 5. How are parametric equations related to finding the equation for a tangent line?

Parametric equations can be used to find the equation for a tangent line by first finding the coordinates of a point on the curve using the parameter(s). Then, the derivative can be used to find the slope of the tangent line at that point. Finally, the point-slope form or slope-intercept form can be used to write the equation of the tangent line.