Reciprocal trig functions and their graphs | Year 11

01 / LEARN

The idea, step by step

Reciprocal is not inverse

Define $sec x = 1 / cos x$ , $csc x = 1 / sin x$ and $cot x = cos x / sin x$ . These are reciprocal functions, not inverse trigonometric functions. Use $arccos$ when necessary to avoid confusing inverse cosine with a reciprocal.

Exclude zeros of the original denominator. Secant is undefined at $π / 2 + k π$ ; cosecant and cotangent are undefined at $k π$ . These points form vertical asymptotes.

sec x = \frac{1}{cos x}, csc x = \frac{1}{sin x}, cot x = \frac{cos x}{sin x}

Sketch separate branches

When cosine is 1 or $- 1$ , secant also equals 1 or $- 1$ . As cosine approaches zero, its reciprocal grows in magnitude. Thus secant has range $(- \infty, - 1] \cup [1, \infty)$ and period $2 π$ ; cosecant has the same range and period.

Cotangent has period $π$ , zeros at $π / 2 + k π$ and range $ℝ$ . Never connect branches across an asymptote. In a reciprocal equation, solve the equivalent sine/cosine equation and check the original exclusions.

A common trap

Using $sec x$ as another notation for inverse cosine.

Instead: Secant is $1 / cos x$ ; inverse cosine returns an angle from an input ratio.

02 / WORKED EXAMPLES

See the thinking

Example 1

State the domain and range of $y = sec x$ .

Exclude every zero of cosine: $x = π / 2 + k π$ .
The reciprocal of a nonzero number between $- 1$ and 1 has magnitude at least 1.
Both endpoints are attained when cosine is $\pm 1$ .

Domain excludes

π / 2 + k π

k \in ℤ

; range

(- \infty, - 1] \cup [1, \infty)

Example 2

Solve $csc x = 2$ on $[0, 2 π)$ .

The original equation requires nonzero sine.
Take reciprocals to obtain $sin x = 1 / 2$ .
The solutions $π / 6, 5 π / 6$ both satisfy the original restriction.

x = π / 6, 5 π / 6

Check your understanding

Where is $cot x = 0$ ?

Show answer

$x = π / 2 + k π$ , $k \in ℤ$ .

Cosine is zero while the sine denominator is nonzero.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Find $sec (π / 3)$ .
2 marks
Find $csc (7 π / 6)$ .
2 marks
State the excluded values for $csc (2 x)$ .
2 marks
Solve $sec x = - 2$ on $[0, 2 π)$ .
2 marks
State the range of $2 sec x + 1$ .
2 marks
Find the period and zeros of $cot (2 x)$ .
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find $sec (π / 3)$ .
2 marks
Take the reciprocal of $cos (π / 3) = 1 / 2$ .
2.
Find $csc (7 π / 6)$ .
2 marks
The sine is $- 1 / 2$ ; its reciprocal is $- 2$ .
-2.
State the excluded values for $csc (2 x)$ .
2 marks
Exclude $sin (2 x) = 0$ , giving $2 x = k π$ .
$x = k π / 2$ , $k \in ℤ$ .
Solve $sec x = - 2$ on $[0, 2 π)$ .
2 marks
Take reciprocals to solve $cos x = - 1 / 2$ .
$x = 2 π / 3, 4 π / 3$ .
State the range of $2 sec x + 1$ .
2 marks
Scale secant by 2 and shift both branches up 1.
$(- \infty, - 1] \cup [3, \infty)$ .
Find the period and zeros of $cot (2 x)$ .
2 marks
Cotangent repeats when its argument increases by $π$ ; its zeros occur at argument $π / 2 + k π$ .
Period $π / 2$ ; zeros $x = π / 4 + k π / 2$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics, Further Trigonometry: periodic functions and models, trigonometric identities and reciprocal functions; the Library sequence follows the approved school-topic structure.

Official curriculum source ↗

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