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Grade 12

Simplify expressions with reciprocals using complementary results

Lesson

In a right-angled triangle, it is clear that $\sin\theta=\cos\left(90^\circ-\theta\right)$sinθ=cos(90°θ) and similarly $\cos\theta=\sin\left(90^\circ-\theta\right)$cosθ=sin(90°θ). This was mentioned in an earlier chapter.

Also, it can be established from the definitions that 

$\cot\theta=\tan\left(90^\circ-\theta\right)$cotθ=tan(90°θ) and $\tan\theta=\cot\left(90^\circ-\theta\right),$tanθ=cot(90°θ),

$\csc\theta=\sec\left(90^\circ-\theta\right)$cscθ=sec(90°θ) and $\sec\theta=\csc\left(90^\circ-\theta\right)$secθ=csc(90°θ)

The trigonometric functions paired in this way are called cofunctions.

The same relations hold for angles of any magnitude. For example, to find the way to re-write $\tan\left(\frac{\pi}{2}-\theta\right)$tan(π2θ) we cannot use the tangent angle difference identity directly because $\tan\frac{\pi}{2}$tanπ2 is undefined. But, we can revert to the definition of the tangent function and write 

$\tan\left(\frac{\pi}{2}-\theta\right)=\frac{\sin\left(\frac{\pi}{2}-\theta\right)}{\cos\left(\frac{\pi}{2}-\theta\right)}=\frac{\sin\frac{\pi}{2}\cos\theta-\cos\frac{\pi}{2}\sin\theta}{\cos\frac{\pi}{2}\cos\theta+\sin\frac{\pi}{2}\sin\theta}$tan(π2θ)=sin(π2θ)cos(π2θ)=sinπ2cosθcosπ2sinθcosπ2cosθ+sinπ2sinθ

Now, we can use the facts that $\sin\frac{\pi}{2}=1$sinπ2=1 and $\cos\frac{\pi}{2}=0$cosπ2=0 to simplify this expression and conclude that 

$\tan\left(\frac{\pi}{2}-\theta\right)=\frac{\cos\theta}{\sin\theta}=\cot\theta$tan(π2θ)=cosθsinθ=cotθ.

The other cofunction identities can be confirmed in a similar way.

Why not see if you can determine them? 

 

Example 1

Rewrite $\sec35^\circ15'$sec35°15 in terms of its cofunction.

The required cofunction is cosecant. We have $\sec35^\circ15'=\csc\left(90^\circ-35^\circ15'\right)$sec35°15=csc(90°35°15). Hence, $\sec35^\circ15'$sec35°15 is the same as $\csc54^\circ45'$csc54°45.

Example 2

Rewrite $\sin0.67$sin0.67 in terms of its cofunction with the argument correct to two decimal places.

The required cofunction is cosine. We have $\sin0.67=\cos\left(\frac{\pi}{2}-0.67\right)$sin0.67=cos(π20.67). Thus, $\sin0.67\approx\cos0.90$sin0.67cos0.90

 

Remember!

Note that in the absence of a $^\circ$° sign, we always assume the angle is given in radians.

More Worked Examples

QUESTION 1

Express $\sin\left(\pi+\theta\right)-\sin\left(\pi-\theta\right)$sin(π+θ)sin(πθ) in simplest form.

QUESTION 2

Rewrite $\sin18^\circ23'$sin18°23 in terms of its cofunction.

QUESTION 3

Rewrite $\sec\frac{\pi}{9}$secπ9 in terms of its cofunction.

Outcomes

12F.B.1.3

Determine, with technology, the primary trigonometric ratios and the reciprocal trigonometric ratios of angles expressed in radian measure

12F.B.3.1

Recognize equivalent trigonometric expressions and verify equivalence using graphing technology

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