Graphing the Tangent Function
Learn to graph y = tan(x), identify asymptotes, period, and key features of the tangent curve.
Definition
- Period: (not like sine and cosine)
- Domain: All real numbers except where is any integer
- Range: All real numbers
- Asymptotes: Vertical lines at
- x-intercepts: At
- The function passes through the origin
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Worked Examples
Sketch one period of and identify the asymptotes, intercepts, and behavior.
Find the asymptotes
when and → Vertical asymptotes at
Find the x-intercept
when , so → x-intercept at
Find key points
and → Points: and
Describe the behavior
As , ; as , → Curve rises from to
Answer: The tangent function has vertical asymptotes at , passes through , and increases continuously from to within each period.
Common Mistakes
Thinking the period of tangent is like sine and cosine
Why it's wrong: The tangent function repeats every radians because for all in the domain.
Correct: The period of is . For , the period is .
Drawing the curve crossing through the asymptotes
Why it's wrong: Asymptotes represent values where the function is undefined. The curve approaches but never touches or crosses them.
Correct: Draw vertical dashed lines for asymptotes. The curve approaches as it nears each asymptote.
Forgetting that tangent has no maximum or minimum value
Why it's wrong: Unlike sine and cosine which are bounded between and , tangent's range is all real numbers.
Correct: The range of is . There is no amplitude for tangent functions.
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Practice Problems
15 problemsWhat is the period of ?
Why It Matters
- Architecture: Calculating roof pitch and ramp angles
- Physics: Analyzing projectile motion and pendulum swings
- Navigation: Determining bearings and flight paths
- Engineering: Designing roads, bridges, and structures with specific inclines
Real World Applications
Road Engineering
Engineers use the tangent function when designing roads with specific gradients and banking angles.
Example:
A road rises 5 meters over a horizontal distance of 100 meters. The grade angle satisfies , giving .
A wheelchair ramp must have a maximum slope ratio of 1:12 (rise to run).
What is the maximum angle of the ramp?
Step 1: Write the mathematical expression
Calculate :
Aviation Navigation
Pilots use tangent ratios when calculating descent angles and approach paths.
Example:
A plane descends from 10,000 feet while traveling 30 miles horizontally. The descent angle has .
A standard 3-degree glide slope is used for landing approaches.
For every mile of horizontal distance, how many feet does the plane descend?
Step 1: Write the mathematical expression
Calculate:
Key Takeaways
- 1The tangent function is defined as
- 2The period of is (not )
- 3Vertical asymptotes occur at where cosine equals zero
- 4The range is all real numbers with no maximum or minimum
- 5For , the period is
- 6The graph passes through the origin and has x-intercepts at multiples of
Frequently Asked Questions
Glossary
- Asymptote
- A line that a curve approaches but never touches. For tangent, these are vertical lines where the function is undefined.
- Period
- The horizontal distance before a function repeats. For , the period is .
- Tangent function
- A trigonometric function defined as , representing the ratio of opposite to adjacent sides in a right triangle.