Tangent and the reciprocal functions

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The problem that opens the lesson

A unit-circle point is $P=(-\frac{\frac{8}{17,15}}{17})$ . 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 $t$ means $\frac{1}{cos}$ t, while arccos is the inverse function of a restricted cosine branch. Following that structure gives $sin=\frac{15}{17,} cos=-\frac{8}{17,} tan=-\frac{15}{8,} csc=\frac{17}{15,} sec=-\frac{17}{8,} cot=-\frac{8}{15}$ .

Why this works

The unit-circle identity produces two more identities by division: $1+tan^2 t=sec^2 t$ 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.

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What this lesson is really about

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

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.

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Why the relationship works

The unit-circle identity produces two more identities by division: $1+tan^2 t=sec^2 t$ and $cot^2 t+1=csc^2 t$ .

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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 $t$ means $\frac{1}{cos}$ 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.

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What commonly goes wrong

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

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.

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Worked examples

Worked example 1

A unit-circle point is $P=(-\frac{\frac{8}{17,15}}{17})$ . 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 $t$ means $\frac{1}{cos}$ t, while arccos is the inverse function of a restricted cosine branch. Following that structure gives $sin=\frac{15}{17,} cos=-\frac{8}{17,} tan=-\frac{15}{8,} csc=\frac{17}{15,} sec=-\frac{17}{8,} cot=-\frac{8}{15}$ .

Why this works

The unit-circle identity produces two more identities by division: $1+tan^2 t=sec^2 t$ 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.

Transfer example

Problem

Derive $1+tan^2 t=sec^2 t$ 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 $t$ means $\frac{1}{cos}$ 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 $[0,2pi)$

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 $t$ means $\frac{1}{cos}$ 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 $t=-\frac{3}{4}$ and $t$ lies in quadrant II, find sin $t$ and cos $t$ .

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 $t$ means $\frac{1}{cos}$ t, while arccos is the inverse function of a restricted cosine branch. Following that structure gives sin $t=\frac{3}{5,}$ cos $t=-\frac{4}{5}$ .

Why this works

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.

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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.

Practice