Dark mode: OFF

cot θ = 7/8 | Find all other trigonometric ratios | Trigonometry Numerical

We have already studied the basics of Trigonometry and the formulas of Trigonometry in the previous notes about Trigonometry. If you haven't read the notes, read it now

Question: If $\ cot \theta = {7 \over 8}$; then find all other trigonometric ratios.

Given that, $cot \theta = {7 \over 8}$

As shown in the figure below, let's assume that we have a right triangle $\triangle ABC$, where $m \angle B = 90 \degree$

Now from the formula of $cot \theta$, that we learned in this notes, we know that:

$cot \ \theta = {\text{Adjacent side to } \theta \over \text{Opposite side to } \theta}$

Putting values in this equation and solving further,

\begin{aligned} \therefore cot \theta &= {BC \over AB} \\ \therefore {7 \over 8} &= {BC \over AB} \\ \end{aligned}

From the equation above, let
$AB = 8 \sdot x$
$BC = 7 \sdot x$

Putting the values of $AB$ and $BC$ in the figure with $\triangle ABC$

Now,
By Pythagoras theorem

\begin{aligned} AC^2 &= AB^2 + BC^2 \\ &= (8x)^2 + (7x)^2 \\ &= 64x^2 + 49x^2 \\ \therefore AC^2 &= 113x^2 \\ \therefore AC &= \sqrt {113x^2} \\ &= \sqrt {113} \sdot x \end{aligned}

Now, that we have the values of $AB$, $BC$ and $AC$ we can find the values of $sin \ \theta$, $cos \ \theta$, $tan \ \theta$, $cosec \ \theta$ and $sec \ \theta$ by applying the formulas that we learnt in the basics of trigonometry

For $sin \ \theta$,

\begin{aligned} sin \ \theta &= {\text {Opposite side to } \theta \over Hypotenuse} \\ \\ &= {AB \over AC} \\ \\ &= {8x \over \sqrt {113} \sdot x} \\ \\ &= {8 \cancel{x} \over \sqrt {113} \sdot \cancel{x}} \\ \\ & \boxed {\therefore sin \ \theta = {8 \over \sqrt {113}}} \\ \end{aligned}

For $cos \ \theta$,

\begin{aligned} cos \ \theta &= {\text {Adjacent side to } \theta \over Hypotenuse} \\ \\ &= {BC \over AC} \\ \\ &= {7x \over \sqrt {113} \sdot x} \\ \\ &= {7 \cancel{x} \over \sqrt {113} \sdot \cancel{x}} \\ \\ & \boxed {\therefore cos \ \theta = {7 \over \sqrt {113}}} \\ \end{aligned}

For $tan \ \theta$,

\begin{aligned} tan \ \theta &= {sin \ \theta \over cos \ \theta} \\ \\ &= {1 \over cot \ \theta} \\ \\ &= {1 \over {7 \over 8}} \\ \\ &= {8 \over 7} \\ \\ & \boxed {\therefore \tan \ \theta = {8 \over 7}} \end{aligned}

For $sec \ \theta$,

\begin{aligned} sec \ \theta &= {1 \over cos \ \theta} \\ \\ &= {1 \over {7 \over \sqrt {113}}} \\ \\ &= {\sqrt {113} \over 7} \\ \\ &\boxed {\therefore sec \ \theta = {\sqrt {113} \over 7}} \end{aligned}

For $cosec \ \theta$,

\begin{aligned} cosec \ \theta &= {1 \over sin \ \theta} \\ \\ &= {1 \over {8 \over \sqrt {113}}} \\ \\ &= {\sqrt {113} \over 8} \\ \\ &\boxed {\therefore cosec \ \theta = {\sqrt {113} \over 8}} \end{aligned}

Help us build Education Lessons

If our notes and videos are helpful to you, kindly support us by making a donation from our support page so we can continue making more content for students like you.

Go to support page