write an equation of the tangent line to y=sec(x) at (pi/3, 2)

TL;DR

The tangent line to the function y=sec(x) at the point (pi/3, 2) can be determined using the derivative and the point-slope form of a line.

Transcript

3.3 number 21 we are going to find an equation of the tangent line to Y is equal to secant x at pi/ 3 comma 2 whenever we're trying to write an equation of a line we need two things first is the slope second it's the point let's focus on the slope first so here we're trying to get the slope of the tangent line slope of a tangent line is exactly the... Read More

Key Insights

• 🫥 The slope of a tangent line is equal to the derivative of the function.
• 🫥 The equation of a tangent line can be determined using the point-slope form.
• 🫥 Evaluating trigonometric values in special triangles can help find the slope of a tangent line.
• 🏙️ The equation of the tangent line to y=sec(x) at pi/3,2 is y = 2√3(x - pi/3) + 2.
• 💁‍♂️ The point-slope form of a line is y - y1 = m(x - x1).
• ❓ Secant(x) is the reciprocal of cosine(x).
• 🙃 In a 30-60-90 triangle, the sides are in the ratio 1:sqrt(3):2.

Questions & Answers

Q: How do we find the slope of the tangent line?

The slope of the tangent line is equal to the derivative of the function. For y=sec(x), the derivative is sec(x) tangent(x).

Q: What is the value of secant(pi/3)?

In the 30-60-90 special triangle, with an angle of pi/3 (or 60 degrees), the ratio of the hypotenuse to the adjacent side is 2:1. Therefore, sec(pi/3) = 2/1 = 2.

Q: What is the formula for the point-slope form of a line?

The point-slope form is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.

Q: How can we convert the equation to a simplified form?

By distributing the 2√3 to the terms inside the parentheses and combining like terms, the equation can be simplified to y = 2√3x - 2√3pi/3 + 2.

Summary & Key Takeaways

• To find the equation of the tangent line, we need to determine the slope and the point. The slope is obtained by taking the derivative of the function, which is sec(x) tangent(x).

• At the point (pi/3, 2), the slope of the tangent line is 2√3.

• Using the slope and the point, the equation of the tangent line is y = 2√3(x - pi/3) + 2.