Reciprocated Trigonometric Functions

Neil Trivedi

Teacher

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Challenging Question

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Neil Trivedi

Maths Lead

Reciprocated Trigonometric Functions

Reciprocated trigonometric functions are derived from the primary trigonometric functions: sine, cosine, and tangent.

Reciprocated Trigonometric Functions

Cosecant (): The reciprocal of sine

Secant (): The reciprocal of cosine

Cotangent (): The reciprocal of tangent

As can also be written as .

Example 1:

Evaluate.

No answer provided.

The graph of is the reciprocal of . At the points where , is undefined, so the graph has vertical asymptotes (i.e., where , in which is an integer.) The graph of also has local maxima and minima at , occurring where .

The graph shows 𝑦=sin𝑥 with marked 𝑥−intercepts, indicating where the vertical asymptotes of the cosecant function 𝑦=csc𝑥 occur.

The graph of 𝑦=csc𝑥 is shown as repeating U-shaped curves between vertical asymptotes at 𝑥=𝑘𝜋, with turning points aligned with the maxima and minima of 𝑦=sin𝑥.
The graph of is the reciprocal of . At the points where , is undefined, so the graph has vertical asymptotes (i.e., where , in which is an odd integer.) Likewise, with the graph, the graph has local minima and maxima at , occurring where .

The graph of 𝑦=cos𝑥 is shown over multiple periods, highlighting that the vertical asymptotes of 𝑦=sec𝑥 occur where cos𝑥=0 (at odd multiples of 𝜋/2).

The graph of 𝑦=sec𝑥 is shown over several periods, with separate branches between vertical asymptotes at 𝑥=𝜋/2+𝑘𝜋, illustrating where sec𝑥 is undefined.

The graph of is the reciprocal of . It looks similar to the graph of , but the key difference is that between the asymptotes, the curves decrease from to , because and has a negative gradient where has a positive one. At the points where , is undefined, so the graph has vertical asymptotes (i.e., where , in which is an integer.)

The graph of 𝑦=tan𝑥 is shown over several periods, with repeating branches between vertical asymptotes at 𝑥=𝜋/2+𝑘𝜋, highlighting where tan𝑥 is undefined.

The graph of 𝑦=cot𝑥 is shown over multiple periods, with decreasing branches between vertical asymptotes at 𝑥=𝑘𝜋, where cot𝑥 is undefined.

Example 2:

Sketch the graph of in the range of .

Step 1: The current range is for . We need to modify it to read . Do this by multiplying everything by . To note, the is outside the angle so it doesn't affect the range.

Step 2: Sketch the graph of for . Vertical asymptotes occur at , where is an integer. The local maxima and minima occur at and , respectively.

Step 3: To obtain the graph of in the range , we un-modify the range by dividing all values (including angles and asymptote positions) from the graph of by . This stretches the graph parallel to the axis by a scale factor of . The values do not change because the transformation of the graph is only occurring along the axis.

Step 4: Finally, apply the , which translates the graph upward by . This gives the graph of in the range . The local maxima and minima also move up by units. The local maximum points, originally at , are now at , while the local minimum points, originally at , are now at .