Math / Trigonometry

Question6) Given that $\sin \frac{\pi}{6}=\frac{1}{2}$ , use an equivalent trigonometric expression to show that $\cos \frac{\pi}{3}=\frac{1}{2}$

Studdy Solution

STEP 1

What is this asking? We're asked to prove that $\cos{\frac{\pi}{3}}$ is equal to $\frac{1}{2}$ , using the fact that $\sin{\frac{\pi}{6}}$ is $\frac{1}{2}$ and some trigonometric magic! Watch out! Remember those tricky trigonometric identities!
We need to pick the right one to connect sine and cosine.

STEP 2

1. Relate the angles
2. Apply the co-function identity
3. Calculate the result

STEP 3

To make it easier to see the relationship between $\frac{\pi}{6}$ and $\frac{\pi}{3}$ , let's convert them to degrees!
Remember, $\pi$ radians is equal to $180$ degrees.
So, $\frac{\pi}{6}$ radians is $\frac{180}{6} = 30$ degrees, and $\frac{\pi}{3}$ radians is $\frac{180}{3} = 60$ degrees.

STEP 4

Notice that $30$ degrees and $60$ degrees are complementary angles, meaning they add up to $90$ degrees ( $30 + 60 = 90$ ).
This is a major clue that we should use a co-function identity!

STEP 5

The co-function identity that relates sine and cosine is $\sin(x) = \cos(90^\circ - x)$ when using degrees, or $\sin(x) = \cos(\frac{\pi}{2} - x)$ when using radians.
This identity tells us that the sine of an angle is equal to the cosine of its complementary angle.

STEP 6

We know that $\sin(\frac{\pi}{6}) = \frac{1}{2}$ .
Let's substitute $x = \frac{\pi}{6}$ into our co-function identity: $\sin(\frac{\pi}{6}) = \cos(\frac{\pi}{2} - \frac{\pi}{6})$ .

STEP 7

Now, let's simplify the expression inside the cosine function: $\frac{\pi}{2} - \frac{\pi}{6} = \frac{3\pi}{6} - \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3}$ .

STEP 8

So, we have $\sin(\frac{\pi}{6}) = \cos(\frac{\pi}{3})$ .
We were given that $\sin(\frac{\pi}{6}) = \frac{1}{2}$ .
Therefore, $\cos(\frac{\pi}{3}) = \frac{1}{2}$ !

STEP 9

We have successfully shown that $\cos{\frac{\pi}{3}} = \frac{1}{2}$ using the given information and the co-function identity!

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Question6) Given that $\sin \frac{\pi}{6}=\frac{1}{2}$ , use an equivalent trigonometric expression to show that $\cos \frac{\pi}{3}=\frac{1}{2}$

Studdy Solution

STEP 1

STEP 2

1. Relate the angles
2. Apply the co-function identity
3. Calculate the result

STEP 3

To make it easier to see the relationship between $\frac{\pi}{6}$ and $\frac{\pi}{3}$ , let's convert them to degrees!
Remember, $\pi$ radians is equal to $180$ degrees.
So, $\frac{\pi}{6}$ radians is $\frac{180}{6} = 30$ degrees, and $\frac{\pi}{3}$ radians is $\frac{180}{3} = 60$ degrees.

STEP 4

Notice that $30$ degrees and $60$ degrees are complementary angles, meaning they add up to $90$ degrees ( $30 + 60 = 90$ ).
This is a major clue that we should use a co-function identity!

STEP 5

The co-function identity that relates sine and cosine is $\sin(x) = \cos(90^\circ - x)$ when using degrees, or $\sin(x) = \cos(\frac{\pi}{2} - x)$ when using radians.
This identity tells us that the sine of an angle is equal to the cosine of its complementary angle.

STEP 6

We know that $\sin(\frac{\pi}{6}) = \frac{1}{2}$ .
Let's substitute $x = \frac{\pi}{6}$ into our co-function identity: $\sin(\frac{\pi}{6}) = \cos(\frac{\pi}{2} - \frac{\pi}{6})$ .

STEP 7

Now, let's simplify the expression inside the cosine function: $\frac{\pi}{2} - \frac{\pi}{6} = \frac{3\pi}{6} - \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3}$ .

STEP 8

So, we have $\sin(\frac{\pi}{6}) = \cos(\frac{\pi}{3})$ .
We were given that $\sin(\frac{\pi}{6}) = \frac{1}{2}$ .
Therefore, $\cos(\frac{\pi}{3}) = \frac{1}{2}$ !

STEP 9

We have successfully shown that $\cos{\frac{\pi}{3}} = \frac{1}{2}$ using the given information and the co-function identity!

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Question6) Given that sin⁡π6=12\sin \frac{\pi}{6}=\frac{1}{2}sin6π​=21​, use an equivalent trigonometric expression to show that cos⁡π3=12\cos \frac{\pi}{3}=\frac{1}{2}cos3π​=21​

Studdy Solution

STEP 1

STEP 2

1. Relate the angles2. Apply the co-function identity3. Calculate the result

STEP 3

STEP 4

Notice that 303030 degrees and 606060 degrees are **complementary angles**, meaning they add up to 909090 degrees (30+60=9030 + 60 = 9030+60=90).This is a major clue that we should use a co-function identity!

STEP 5

STEP 6

We know that sin⁡(π6)=12\sin(\frac{\pi}{6}) = \frac{1}{2}sin(6π​)=21​.Let's substitute x=π6x = \frac{\pi}{6}x=6π​ into our co-function identity: sin⁡(π6)=cos⁡(π2−π6)\sin(\frac{\pi}{6}) = \cos(\frac{\pi}{2} - \frac{\pi}{6})sin(6π​)=cos(2π​−6π​).

STEP 7

Now, let's simplify the expression inside the cosine function: π2−π6=3π6−π6=2π6=π3\frac{\pi}{2} - \frac{\pi}{6} = \frac{3\pi}{6} - \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3}2π​−6π​=63π​−6π​=62π​=3π​.

STEP 8

So, we have sin⁡(π6)=cos⁡(π3)\sin(\frac{\pi}{6}) = \cos(\frac{\pi}{3})sin(6π​)=cos(3π​).We were given that sin⁡(π6)=12\sin(\frac{\pi}{6}) = \frac{1}{2}sin(6π​)=21​.Therefore, cos⁡(π3)=12\cos(\frac{\pi}{3}) = \frac{1}{2}cos(3π​)=21​!

STEP 9

We have successfully shown that cos⁡π3=12\cos{\frac{\pi}{3}} = \frac{1}{2}cos3π​=21​ using the given information and the co-function identity!

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Question6) Given that $\sin \frac{\pi}{6}=\frac{1}{2}$ , use an equivalent trigonometric expression to show that $\cos \frac{\pi}{3}=\frac{1}{2}$

1. Relate the angles
2. Apply the co-function identity
3. Calculate the result

Notice that $30$ degrees and $60$ degrees are complementary angles, meaning they add up to $90$ degrees ( $30 + 60 = 90$ ).
This is a major clue that we should use a co-function identity!

We know that $\sin(\frac{\pi}{6}) = \frac{1}{2}$ .
Let's substitute $x = \frac{\pi}{6}$ into our co-function identity: $\sin(\frac{\pi}{6}) = \cos(\frac{\pi}{2} - \frac{\pi}{6})$ .

Now, let's simplify the expression inside the cosine function: $\frac{\pi}{2} - \frac{\pi}{6} = \frac{3\pi}{6} - \frac{\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3}$ .

So, we have $\sin(\frac{\pi}{6}) = \cos(\frac{\pi}{3})$ .
We were given that $\sin(\frac{\pi}{6}) = \frac{1}{2}$ .
Therefore, $\cos(\frac{\pi}{3}) = \frac{1}{2}$ !

We have successfully shown that $\cos{\frac{\pi}{3}} = \frac{1}{2}$ using the given information and the co-function identity!