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BetterGrades Precalculus · Unit 9 · Lesson

Tangent and the reciprocal functions

Define tangent, cotangent, secant, and cosecant from sine and cosine and state domains, signs, periods, and identities.

Textbook reading

The problem that opens the lesson

A unit-circle point is P=(−817,1517)P=(-\frac{\frac{8}{17,15}}{17}). Find all six trig values and identify which are negative.

Solution

Begin by identifying the mathematical object and the information that fixes it. Start with sine and cosine coordinates, form ratios and reciprocals, simplify exact radicals, and retain undefined cases. The relevant conditions are not optional bookkeeping: A reciprocal trig function is not an inverse trig function. The notation sec tt means 1cos\frac{1}{cos} t, while arccos is the inverse function of a restricted cosine branch. Following that structure gives sin=1517,cos=−817,tan=−158,csc=1715,sec=−178,cot=−815sin=\frac{15}{17,} cos=-\frac{8}{17,} tan=-\frac{15}{8,} csc=\frac{17}{15,} sec=-\frac{17}{8,} cot=-\frac{8}{15}.

Why this works

The unit-circle identity produces two more identities by division: 1+tan2t=sec2t1+tan^2 t=sec^2 t and cot2t+1=csc2tcot^2 t+1=csc^2 t. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Textbook reading

What this lesson is really about

Tangent and cotangent are coordinate ratios, while secant and cosecant are reciprocals: tan=sincos,cot=cossin,sec=1cos,csc=1sintan=\frac{sin}{cos}, cot=\frac{cos}{sin}, sec=\frac{1}{cos}, csc=\frac{1}{sin}.

The definitions make domain restrictions visible. Tangent and secant are undefined where cosine is zero; cotangent and cosecant are undefined where sine is zero. Tangent and cotangent repeat after pi because both numerator and denominator change sign after a half-turn.

The point is not merely to reproduce a formula. A learner should be able to identify the quantities or geometric objects involved, explain why the relationship has its stated form, and recognize when the same idea appears in a graph, table, diagram, or model.

Textbook reading

Why the relationship works

The unit-circle identity produces two more identities by division: 1+tan2t=sec2t1+tan^2 t=sec^2 t and cot2t+1=csc2tcot^2 t+1=csc^2 t.

Textbook reading

A reliable way to work

Start with sine and cosine coordinates, form ratios and reciprocals, simplify exact radicals, and retain undefined cases.

A reciprocal trig function is not an inverse trig function. The notation sec tt means 1cos\frac{1}{cos} t, while arccos is the inverse function of a restricted cosine branch.

After the symbolic work is complete, check the result. Depending on the lesson, this may mean substituting into an original equation, comparing coordinates, examining a graph, checking units, testing an interval, or confirming that every branch of a periodic solution has been included.

Textbook reading

What commonly goes wrong

A common error is to invert an angle or to write tan tt as cossin\frac{cos}{sin}.

The repair is to return to the definition and identify the first step where the invalid solution stops describing the original mathematical object. Later algebra cannot rescue a first step that changed the domain, orientation, branch, or meaning of the problem.

Textbook reading

Worked examples

Worked example 1

A unit-circle point is P=(−817,1517)P=(-\frac{\frac{8}{17,15}}{17}). Find all six trig values and identify which are negative.

Solution

Begin by identifying the mathematical object and the information that fixes it. Start with sine and cosine coordinates, form ratios and reciprocals, simplify exact radicals, and retain undefined cases. The relevant conditions are not optional bookkeeping: A reciprocal trig function is not an inverse trig function. The notation sec tt means 1cos\frac{1}{cos} t, while arccos is the inverse function of a restricted cosine branch. Following that structure gives sin=1517,cos=−817,tan=−158,csc=1715,sec=−178,cot=−815sin=\frac{15}{17,} cos=-\frac{8}{17,} tan=-\frac{15}{8,} csc=\frac{17}{15,} sec=-\frac{17}{8,} cot=-\frac{8}{15}.

Why this works

The unit-circle identity produces two more identities by division: 1+tan2t=sec2t1+tan^2 t=sec^2 t and cot2t+1=csc2tcot^2 t+1=csc^2 t. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Transfer example

Problem

Derive 1+tan2t=sec2t1+tan^2 t=sec^2 t from the unit-circle identity.

Worked development

Start with sine and cosine coordinates, form ratios and reciprocals, simplify exact radicals, and retain undefined cases. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. The definitions make domain restrictions visible. Tangent and secant are undefined where cosine is zero; cotangent and cosecant are undefined where sine is zero. Tangent and cotangent repeat after pi because both numerator and denominator change sign after a half-turn. Then apply the conditions explicitly: A reciprocal trig function is not an inverse trig function. The notation sec tt means 1cos\frac{1}{cos} t, while arccos is the inverse function of a restricted cosine branch. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

The six functions provide the language for slope, projection, triangles, identities, and periodic graphs.

Reasoning example

Problem

Find where tangent is undefined on[0,2pi)[0,2pi)

Worked development

Start with sine and cosine coordinates, form ratios and reciprocals, simplify exact radicals, and retain undefined cases. In this example, the first useful move is to make the defining structure visible rather than to search for a memorized answer. The definitions make domain restrictions visible. Tangent and secant are undefined where cosine is zero; cotangent and cosecant are undefined where sine is zero. Tangent and cotangent repeat after pi because both numerator and denominator change sign after a half-turn. Then apply the conditions explicitly: A reciprocal trig function is not an inverse trig function. The notation sec tt means 1cos\frac{1}{cos} t, while arccos is the inverse function of a restricted cosine branch. Finish by checking the result in a second representation and explaining what the result means.

Interpretation

The six functions provide the language for slope, projection, triangles, identities, and periodic graphs.

Worked example 4: quick check

If tan t=−34t=-\frac{3}{4} and tt lies in quadrant II, find sin tt and cos tt.

Solution

Begin by identifying the mathematical object and the information that fixes it. Start with sine and cosine coordinates, form ratios and reciprocals, simplify exact radicals, and retain undefined cases. The relevant conditions are not optional bookkeeping: A reciprocal trig function is not an inverse trig function. The notation sec tt means 1cos\frac{1}{cos} t, while arccos is the inverse function of a restricted cosine branch. Following that structure gives sin t=35,t=\frac{3}{5,} cos t=−45t=-\frac{4}{5}.

Why this works

The unit-circle identity produces two more identities by division: 1+tan2t=sec2t1+tan^2 t=sec^2 t and cot2t+1=csc2tcot^2 t+1=csc^2 t. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Six-function ratio web. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The unit-circle identity produces two more identities by division: 1+tan^2 t=sec^2 t and cot^2 t+1=csc^2 t. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.
Read this graph as text

Tangent and the reciprocal functions · Six-function ratio web. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The unit-circle identity produces two more identities by division: 1+tan^2 t=sec^2 t and cot^2 t+1=csc^2 t. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.

Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.

Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Define tangent, cotangent, secant, and cosecant from sine and cosine and state domains, signs, periods, and identities.

Anchor figure · Six-function ratio web

Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The unit-circle identity produces two more identities by division: 1+tan2t=sec2t1+tan^2 t=sec^2 t and cot2t+1=csc2tcot^2 t+1=csc^2 t. The calculation and the representation should agree, so a graph, diagram, table, or substitution check should support the same conclusion.

Pythagorean identity derivation by division. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for tangent and the reciprocal functions.
Read this graph as text

Tangent and the reciprocal functions · Pythagorean identity derivation by division. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for tangent and the reciprocal functions. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.

Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.

Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Define tangent, cotangent, secant, and cosecant from sine and cosine and state domains, signs, periods, and identities.

Mechanism figure · Pythagorean identity derivation by division

Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for tangent and the reciprocal functions.

Domain-exclusion unit circle with zero coordinates marked. Compare the valid path with the tempting shortcut. The figure shows why to invert an angle or to write tan t as cos/sin leads to a false conclusion.
Read this graph as text

Tangent and the reciprocal functions · Domain-exclusion unit circle with zero coordinates marked. Compare the valid path with the tempting shortcut. The figure shows why to invert an angle or to write tan t as cos/sin leads to a false conclusion. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.

Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.

Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Define tangent, cotangent, secant, and cosecant from sine and cosine and state domains, signs, periods, and identities.

Comparison and error figure · Domain-exclusion unit circle with zero coordinates marked

Compare the valid path with the tempting shortcut. The figure shows why to invert an angle or to write tan tt as cossin\frac{cos}{sin} leads to a false conclusion.

Textbook reading

Application and interpretation

The six functions provide the language for slope, projection, triangles, identities, and periodic graphs.

A contextual answer must include units, a meaningful domain, and the assumptions that make the model plausible. An exact mathematical relationship should not be diluted into a decimal unless a measurement or comparison requires it.

Check yourself

If tan t=−34t=-\frac{3}{4} and tt lies in quadrant II, find sin tt and cos tt.

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Practice

16 concrete questions

Practice 1 · retrieval · foundational01

If tan t=−34t=-\frac{3}{4} and tt lies in quadrant II, find sin tt and cos tt.

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Practice 2 · conceptual · foundational02

State the defining idea behind tangent and the reciprocal functions in one precise sentence.

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Practice 3 · verification · developing03

For tangent and the reciprocal functions, what condition or domain restriction must remain visible in the solution?

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Practice 4 · error analysis · developing04

For tangent and the reciprocal functions, describe the most likely incorrect first step and explain why it fails.

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Practice 5 · synthesis · transfer05

For tangent and the reciprocal functions, explain how this lesson's idea will be used later in the course.

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Practice 6 · procedural · foundational06

Solve this tangent and the reciprocal functions problem and state the final result: A unit-circle point is P=(−817,1517)P=(-\frac{\frac{8}{17,15}}{17}). Find all six trig values and identify which are negative.

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Practice 7 · procedural · developing07

In tangent and the reciprocal functions, for “Derive 1+tan2t=sec2t1+tan^2 t=sec^2 t from the unit-circle identity.”, identify the first valid mathematical step and the condition that must remain visible.

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Practice 8 · transfer · transfer08

For “Find where tangent is undefined on [0,2pi).”,[0,2pi).”, identify the governing definition or relationship and what a complete conclusion must include.

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Practice 9 · verification · developing09

Verify “sin=1517,cos=−817,tan=−158,csc=1715,sec=−178,cot=−815sin=\frac{15}{17,} cos=-\frac{8}{17,} tan=-\frac{15}{8,} csc=\frac{17}{15,} sec=-\frac{17}{8,} cot=-\frac{8}{15}.” using the required condition for tangent and the reciprocal functions.

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Practice 10 · explanation · developing10

Explain why “sin=1517,cos=−817,tan=−158,csc=1715,sec=−178,cot=−815sin=\frac{15}{17,} cos=-\frac{8}{17,} tan=-\frac{15}{8,} csc=\frac{17}{15,} sec=-\frac{17}{8,} cot=-\frac{8}{15}.” follows from this lesson’s mathematical mechanism.

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Practice 11 · conceptual · developing11

What mathematical structure is shared by the opening problem and “Find where tangent is undefined on [0,2pi)[0,2pi).”?

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Practice 12 · graphical · developing12

In “Six-function ratio web”, which mathematical objects or labels must be visible to support “sin=1517,cos=−817,tan=−158,csc=1715,sec=−178,cot=−815sin=\frac{15}{17,} cos=-\frac{8}{17,} tan=-\frac{15}{8,} csc=\frac{17}{15,} sec=-\frac{17}{8,} cot=-\frac{8}{15}.”?

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Practice 13 · graphical · transfer13

How should “Pythagorean identity derivation by division” make the governing relationship in “Derive 1+tan2t=sec2t1+tan^2 t=sec^2 t from the unit-circle identity.” visible?

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Practice 14 · error analysis · transfer14

In “Domain-exclusion unit circle with zero coordinates marked”, identify the first point where the misconception diverges from valid tangent and the reciprocal functions reasoning.

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Practice 15 · modeling · transfer15

In the application “The six functions provide the language for slope, projection, triangles, identities, and periodic graphs.”, what quantities or geometric objects must be identified, and what condition makes the model valid?

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Practice 16 · exit check · transfer16

Answer “If tan t=−34t=-\frac{3}{4} and tt lies in quadrant II, find sin tt and cos tt.” and name the condition used to check the result.

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Textbook reading

Lesson summary

Tangent and cotangent are coordinate ratios, while secant and cosecant are reciprocals: tan=sincos,cot=cossin,sec=1cos,csc=1sintan=\frac{sin}{cos}, cot=\frac{cos}{sin}, sec=\frac{1}{cos}, csc=\frac{1}{sin}.

The central condition to remember is this: A reciprocal trig function is not an inverse trig function. The notation sec tt means 1cos\frac{1}{cos} t, while arccos is the inverse function of a restricted cosine branch.

Connection forward

The next unit unwraps these coordinate functions into graphs over the real line.

The next lesson is Building the sine graph from circular motion.

Source record

Original BetterGrades manuscript, rights-separated references.

  • Sundstrom & Schlicker, Trigonometry 1.1-1.6
  • Lippman & Rasmussen, Precalculus Vol. 2, 5.1-5.4
  • Yoshiwara, Trigonometry, Chapters 4 and 6
  • Corral, Trigonometry, Chapter 4
  • Stitz & Zeager, Precalculus, Chapter 10

No long source passage is reproduced.