QuestionIdentify the invalid equation from the options below: A. $\sin \frac{\pi}{3}=\cos \frac{\pi}{6}$ B. $\sin \frac{\pi}{4}=\cos \frac{\pi}{4}$ C. $\csc \frac{\pi}{6}=\cos \frac{\pi}{3}$ D. $\tan \frac{\pi}{4}=\cot \frac{\pi}{4}$

Studdy Solution

STEP 1

Assumptions1. We are working in the domain of real numbers. . We are using the unit circle definition of trigonometric functions.
3. We are using radians as the measure of angles.
4. We know the values of sine, cosine, tangent, and their reciprocal functions at common angles.

STEP 2

We will evaluate each option one by one. Let's start with option A. We know that $\sin \frac{\pi}{}=\frac{\sqrt{}}{2}$ and $\cos \frac{\pi}{6}=\frac{\sqrt{}}{2}$ .

STEP 3

Since both sides of the equation in option A are equal, option A is a valid equation.

STEP 4

Now, let's evaluate option B. We know that $\sin \frac{\pi}{4}=\frac{\sqrt{2}}{2}$ and $\cos \frac{\pi}{4}=\frac{\sqrt{2}}{2}$ .

STEP 5

Since both sides of the equation in option B are equal, option B is a valid equation.

STEP 6

Now, let's evaluate option C. We know that $\csc \frac{\pi}{6}=2$ and $\cos \frac{\pi}{3}=\frac{1}{2}$ .

STEP 7

Since both sides of the equation in option C are not equal, option C is not a valid equation.

STEP 8

We could stop here since we have found the invalid equation. But for completeness, let's evaluate option D. We know that $\tan \frac{\pi}{4}=1$ and $\cot \frac{\pi}{4}=1$ .

STEP 9

Since both sides of the equation in option D are equal, option D is a valid equation.
The invalid equation is option C $\csc \frac{\pi}{6}=\cos \frac{\pi}{3}$ .

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QuestionIdentify the invalid equation from the options below: A. $\sin \frac{\pi}{3}=\cos \frac{\pi}{6}$ B. $\sin \frac{\pi}{4}=\cos \frac{\pi}{4}$ C. $\csc \frac{\pi}{6}=\cos \frac{\pi}{3}$ D. $\tan \frac{\pi}{4}=\cot \frac{\pi}{4}$

Studdy Solution

STEP 1

Assumptions1. We are working in the domain of real numbers. . We are using the unit circle definition of trigonometric functions.
3. We are using radians as the measure of angles.
4. We know the values of sine, cosine, tangent, and their reciprocal functions at common angles.

STEP 2

We will evaluate each option one by one. Let's start with option A. We know that $\sin \frac{\pi}{}=\frac{\sqrt{}}{2}$ and $\cos \frac{\pi}{6}=\frac{\sqrt{}}{2}$ .

STEP 3

Since both sides of the equation in option A are equal, option A is a valid equation.

STEP 4

Now, let's evaluate option B. We know that $\sin \frac{\pi}{4}=\frac{\sqrt{2}}{2}$ and $\cos \frac{\pi}{4}=\frac{\sqrt{2}}{2}$ .

STEP 5

Since both sides of the equation in option B are equal, option B is a valid equation.

STEP 6

Now, let's evaluate option C. We know that $\csc \frac{\pi}{6}=2$ and $\cos \frac{\pi}{3}=\frac{1}{2}$ .

STEP 7

Since both sides of the equation in option C are not equal, option C is not a valid equation.

STEP 8

We could stop here since we have found the invalid equation. But for completeness, let's evaluate option D. We know that $\tan \frac{\pi}{4}=1$ and $\cot \frac{\pi}{4}=1$ .

STEP 9

Since both sides of the equation in option D are equal, option D is a valid equation.
The invalid equation is option C $\csc \frac{\pi}{6}=\cos \frac{\pi}{3}$ .

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Studdy Solution

STEP 1

Assumptions1. We are working in the domain of real numbers. . We are using the unit circle definition of trigonometric functions.3. We are using radians as the measure of angles.4. We know the values of sine, cosine, tangent, and their reciprocal functions at common angles.

STEP 2

We will evaluate each option one by one. Let's start with option A. We know that sin⁡π=2\sin \frac{\pi}{}=\frac{\sqrt{}}{2}sinπ​=2​​ and cos⁡π6=2\cos \frac{\pi}{6}=\frac{\sqrt{}}{2}cos6π​=2​​.

STEP 3

Since both sides of the equation in option A are equal, option A is a valid equation.

STEP 4

Now, let's evaluate option B. We know that sin⁡π4=22\sin \frac{\pi}{4}=\frac{\sqrt{2}}{2}sin4π​=22​​ and cos⁡π4=22\cos \frac{\pi}{4}=\frac{\sqrt{2}}{2}cos4π​=22​​.

STEP 5

Since both sides of the equation in option B are equal, option B is a valid equation.

STEP 6

Now, let's evaluate option C. We know that csc⁡π6=2\csc \frac{\pi}{6}=2csc6π​=2 and cos⁡π3=12\cos \frac{\pi}{3}=\frac{1}{2}cos3π​=21​.

STEP 7

Since both sides of the equation in option C are not equal, option C is not a valid equation.

STEP 8

We could stop here since we have found the invalid equation. But for completeness, let's evaluate option D. We know that tan⁡π4=1\tan \frac{\pi}{4}=1tan4π​=1 and cot⁡π4=1\cot \frac{\pi}{4}=1cot4π​=1.

STEP 9

Since both sides of the equation in option D are equal, option D is a valid equation.The invalid equation is option C csc⁡π6=cos⁡π3\csc \frac{\pi}{6}=\cos \frac{\pi}{3}csc6π​=cos3π​.

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Assumptions1. We are working in the domain of real numbers. . We are using the unit circle definition of trigonometric functions.
3. We are using radians as the measure of angles.
4. We know the values of sine, cosine, tangent, and their reciprocal functions at common angles.

We will evaluate each option one by one. Let's start with option A. We know that $\sin \frac{\pi}{}=\frac{\sqrt{}}{2}$ and $\cos \frac{\pi}{6}=\frac{\sqrt{}}{2}$ .

Now, let's evaluate option B. We know that $\sin \frac{\pi}{4}=\frac{\sqrt{2}}{2}$ and $\cos \frac{\pi}{4}=\frac{\sqrt{2}}{2}$ .

Now, let's evaluate option C. We know that $\csc \frac{\pi}{6}=2$ and $\cos \frac{\pi}{3}=\frac{1}{2}$ .

We could stop here since we have found the invalid equation. But for completeness, let's evaluate option D. We know that $\tan \frac{\pi}{4}=1$ and $\cot \frac{\pi}{4}=1$ .

Since both sides of the equation in option D are equal, option D is a valid equation.
The invalid equation is option C $\csc \frac{\pi}{6}=\cos \frac{\pi}{3}$ .