How to find the exact values of cot, csc and sec for 45 degrees?

Question

Olivia Gallegos · Accepted Answer

Recall what each of the reciprocal trig identities means.

\cot x = \frac{1}{\tan x} \frac{2}{2} \csc x = \frac{1}{\sin x} \frac{2}{2} \sec x = \frac{1}{\cos x}

On the unit circle, the coordinates for

45^{\circ}

are

(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})

Where the

x

coordinate is the

\cos

value, and the

y

coordinate is the

\sin

value.

\tan x

is defined as

\frac{\sin x}{\cos x}

. With the use of this data, we can

\tan x = 1 \frac{2}{2} \sin x = \frac{\sqrt{2}}{2} \frac{2}{2} \cos x = \frac{\sqrt{2}}{2}

These values can be plugged into our equations to obtain the reciprocal identities.

\cot x = \frac{1}{1} = 1 \frac{2}{2} \csc x = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2} \frac{2}{2} \sec x = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}

Remember for

{\csc 45}^{\circ}

and

{\sec 45}^{\circ}

, we had to rationalize the denominator.

\bar{\underset{̲}{| \frac{2}{2} {\cot 45}^{\circ} = 1 \frac{2}{2} {\csc 45}^{\circ} = \sqrt{2} \frac{2}{2} {\sec 45}^{\circ} = \sqrt{2} \frac{2}{2} |}}

onteringsdh · Accepted Answer

Draw a             45      ∘       right triangle. A             45      ∘       right triangle has 2 equal sides plus the hypotenuse, right? If you give the 2 equal sides a length of 1 unit, Pythagoras will tell you that the hypotenuse has a length of                     2            . Label the angles&#039; degrees as well as the lengths of the sides. Do not lose this drawing.You should have these memorized:sine is opposite/hypotenusecosine is adjacent/hypotenusetangent is opposite/adjacentThis inquiry requests the cot, csc, and sec. Also, commit the following information to memory:cosecant is 1/sinesecant is 1/cosinecotangent is 1/tangentConsidering the bolded information above,                              cot          45                ∘            =              adjacent/opposite                                    csc          45                ∘            =              hypotenuse/opposite                                    sec          45                ∘            =              hypotenuse/adjacent      The lengths indicated in the aforementioned fractions for each trig function should now be plugged into your drawing of the             45      ∘       right triangle.

How to find the exact values of cot, csc and sec for 45 degrees?

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