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Math  /  Trigonometry

Question13) sec⁡45∘\sec 45^{\circ}

Studdy Solution

STEP 1

What is this asking? What's the secant of 45 degrees? Watch out! Don't mix up sine, cosine, tangent, secant, cosecant, and cotangent!

STEP 2

1. Relate secant to cosine.
2. Evaluate the cosine.
3. Calculate the secant.

STEP 3

Alright, let's **kick things off**!
We're looking for the secant of 45 degrees.
Remember that secant is the reciprocal of cosine.
So, we can write this as: sec⁡(45∘)=1cos⁡(45∘) \sec(45^{\circ}) = \frac{1}{\cos(45^{\circ})} This is a **key relationship** to remember!

STEP 4

Now, what's the cosine of 45 degrees?
Think back to your **unit circle** or your **special right triangles**!
We know that: cos⁡(45∘)=22 \cos(45^{\circ}) = \frac{\sqrt{2}}{2} This is a **fundamental value** that we should definitely know!

STEP 5

Now we can **plug this value back** into our expression for the secant: sec⁡(45∘)=122 \sec(45^{\circ}) = \frac{1}{\frac{\sqrt{2}}{2}}

STEP 6

Dividing by a fraction is the same as multiplying by its reciprocal.
So, we **flip the fraction** in the denominator and multiply: sec⁡(45∘)=1⋅22=22 \sec(45^{\circ}) = 1 \cdot \frac{2}{\sqrt{2}} = \frac{2}{\sqrt{2}}

STEP 7

We usually like to **rationalize the denominator**, which means getting rid of any square roots on the bottom.
We can do this by multiplying both the numerator and denominator by 2\sqrt{2}: sec⁡(45∘)=22⋅22=2⋅22 \sec(45^{\circ}) = \frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\cdot \sqrt{2}}{2}

STEP 8

Now, we can **simplify** by dividing the numerator and denominator by 2\textbf{2}: sec⁡(45∘)=2⋅22=2 \sec(45^{\circ}) = \frac{2\cdot \sqrt{2}}{2} = \sqrt{2} And there you have it!

STEP 9

The secant of 45 degrees is 2\sqrt{2}.

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