- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 3
Exponential and trigonometric functions practice question 9
Find the -intercept of the line tangent to at .
The intercept belongs to the tangent line, so first you need its point and slope at . Which exact values of sine and cosine will come up?
The slope is , and the point is .
Write the line in point-slope form and set .
An exact expression, or a decimal to two decimal places.
Theory UBC Math 100 §4.2CLP-1 §2.8Key results
Worked solution
Strategy
Point plus slope gives the tangent line; the intercept is its value at x = 0. Every trigonometric value at π is one you know exactly.
Working
Point (π, 3π), slope 2.
Point-slope form, simplified.
Set x = 0.
Answer
The answer is pi, about 3.14.
Check
Near x = π, sin x ≈ π − x, so the curve is close to (π − x) + 3x = 2x + π, a line with intercept π.
Where this goes wrong
Taking cos π = 1 and getting slope 4, or sin π = 1 and getting the point wrong. At π the cosine is −1 and the sine is 0.
Leans on
Values of sin and cos at π, from Functions, graphs and inverses
Try a similar question
Find an equation of the tangent line to at the point where .Saved in this browser only.