5.3Exploring Tangent and the Reciprocal Functions
Define tan θ = sin/cos, cot θ = cos/sin, sec θ = 1/cos, csc θ = 1/sin. Evaluate all six trig functions from the unit circle and identify where each is undefined.
Tangent, secant, cosecant, and cotangent complete the six trig functions. All six appear in calculus derivatives and integrals — knowing their definitions as ratios of sine and cosine makes every trig identity and derivative derivable.
Essential Question
How do tangent, cotangent, secant, and cosecant extend the two basic trig functions — and where do they break down?
Lesson Overview
Sine and cosine describe the y- and x-coordinates of a point on the unit circle. The other four trig functions are built entirely from those two ratios. Tangent is the slope of the terminal side; cotangent is its reciprocal. Secant and cosecant are the reciprocals of cosine and sine. Because they involve division, each function is undefined whenever its denominator equals zero — producing vertical asymptotes in their graphs.
All Six Trig Functions at a Glance
Geometric Meaning of Tangent
When the unit circle is drawn with a vertical tangent line at (1, 0), the length of the segment from (1, 0) to where the terminal side meets that line equals tan θ.
Where Each Function Is Undefined
Pythagorean Identities
Divide sin²θ + cos²θ = 1 by cos²θ to get the tangent form; divide by sin²θ to get the cotangent form.
Signs in Each Quadrant
Key Vocabulary
Tangent (tan θ)
tan θ = sin θ / cos θ = y/x. Undefined when cos θ = 0 (i.e., at π/2 + nπ).
Example: tan(π/4) = 1
Cotangent (cot θ)
cot θ = cos θ / sin θ = x/y. Undefined when sin θ = 0 (i.e., at nπ).
Example: cot(π/4) = 1
Secant (sec θ)
sec θ = 1 / cos θ = r/x. Undefined when cos θ = 0.
Example: sec(π/3) = 2
Cosecant (csc θ)
csc θ = 1 / sin θ = r/y. Undefined when sin θ = 0.
Example: csc(π/6) = 2
Reciprocal identity
Each of csc, sec, cot is the reciprocal of sin, cos, tan respectively.
Quotient identity
tan θ = sin θ / cos θ and cot θ = cos θ / sin θ express tan and cot as quotients.
Worked Examples
Find all six trig functions for the point (3/5, 4/5) on the unit circle.
Since the point is on the unit circle, r = 1, x = 3/5, y = 4/5.
sin θ = y = 4/5, cos θ = x = 3/5
tan θ = y/x = (4/5)/(3/5) = 4/3
csc θ = 1/sin θ = 5/4, sec θ = 1/cos θ = 5/3
cot θ = 1/tan θ = 3/4
Evaluate all six trig functions at θ = π/6 (30°).
Unit circle: (cos π/6, sin π/6) = (√3/2, 1/2)
sin π/6 = 1/2, cos π/6 = √3/2
tan π/6 = (1/2)/(√3/2) = 1/√3 = √3/3
csc π/6 = 2, sec π/6 = 2/√3 = 2√3/3
cot π/6 = √3
Use a Pythagorean identity: if tan θ = 2 and θ is in QI, find sec θ.
Identity: 1 + tan²θ = sec²θ
1 + (2)² = sec²θ
1 + 4 = sec²θ → sec²θ = 5
sec θ = √5 (positive because θ is in QI)
Determine the sign of cot(5π/4).
5π/4 is in QIII (between π and 3π/2).
cot θ = cos θ / sin θ = (−)/(−) = positive.
Find csc θ if sin θ = −√2/2 and θ is in QIII.
csc θ = 1 / sin θ
csc θ = 1 / (−√2/2) = −2/√2 = −√2
Guided Practice
The point (−5/13, 12/13) is on the unit circle. Find tan θ and cot θ.
Hint: tan θ = y/x. Substitute x = −5/13 and y = 12/13 directly.
Evaluate sec(π/4) and csc(π/4).
Hint: cos(π/4) = sin(π/4) = √2/2. Take reciprocals.
If cot θ = −3/4 and sin θ {'>'} 0, find csc θ using 1 + cot²θ = csc²θ.
Hint: Square cot θ, add 1, then take the square root. Use the sign of sin θ to choose + or −.
In which quadrant is θ if tan θ {'<'} 0 and sec θ {'>'} 0?
Hint: sec θ {'>'} 0 means cos θ {'>'} 0. tan θ {'<'} 0 with cos θ {'>'} 0 means sin θ {'<'} 0. Which quadrant has cos + and sin −?
Evaluate tan(3π/2). Is it defined? Explain.
Hint: cos(3π/2) = 0. What happens when you divide by zero?
Check Your Understanding
Interactive Practice — 5 Questions
Which expression equals tan θ?
At which angle is tan θ undefined?
Which Pythagorean identity involves sec θ?
In QII, which of the following is positive?
If cos θ = 1/3, what is sec θ?
Common Mistakes
Writing csc θ = 1/tan θ (confusing cosecant with cotangent).
csc θ = 1/sin θ and cot θ = 1/tan θ. Cosecant is the reciprocal of sine.
Saying tan(π/2) = 0 because "it looks like a right angle."
tan(π/2) is undefined — cos(π/2) = 0 and division by zero is undefined.
Forgetting to check the quadrant when taking a square root in a Pythagorean identity.
After sec²θ = 5, decide sec θ = +√5 or −√5 based on which quadrant θ is in.
Math Tips
The reciprocal pairs are: sin↔csc, cos↔sec, tan↔cot. Their product always equals 1.
tan θ is the slope of the terminal side of angle θ in standard position.
Period of tan and cot is π. Period of sec and csc is 2π — same as sin and cos.
Mnemonic for positive functions by quadrant: "All Students Take Calculus" (QI→QII→QIII→QIV).