Math / Trigonometry

Question1. a. Express $\cos (x)$ in terms of $\cot (x)$ and $\csc (x)$ . b. Express $\csc (x)$ in terms of $\cos (x)$ and $\tan (x)$ .

Studdy Solution

STEP 1

1. We are working with trigonometric identities and relationships.
2. We will use basic trigonometric identities to express one function in terms of others.
3. We assume $x$ is an angle where all trigonometric functions are defined.

STEP 2

1. Express $\cos(x)$ in terms of $\cot(x)$ and $\csc(x)$ .
2. Express $\csc(x)$ in terms of $\cos(x)$ and $\tan(x)$ .

STEP 3

Start with the identity for $\cot(x)$ and $\csc(x)$ :
$\cot(x) = \frac{\cos(x)}{\sin(x)}$
$\csc(x) = \frac{1}{\sin(x)}$

STEP 4

Solve the expression for $\cos(x)$ in terms of $\cot(x)$ and $\csc(x)$ :
Since $\cot(x) = \frac{\cos(x)}{\sin(x)}$ , we can express $\sin(x)$ in terms of $\csc(x)$ :
$\sin(x) = \frac{1}{\csc(x)}$
Substitute $\sin(x)$ into the expression for $\cot(x)$ :
$\cot(x) = \frac{\cos(x)}{\frac{1}{\csc(x)}} = \cos(x) \cdot \csc(x)$
Thus, $\cos(x) = \cot(x) \cdot \frac{1}{\csc(x)} = \frac{\cot(x)}{\csc(x)}$ .

STEP 5

Start with the identity for $\tan(x)$ and $\cos(x)$ :
$\tan(x) = \frac{\sin(x)}{\cos(x)}$

STEP 6

Express $\sin(x)$ in terms of $\cos(x)$ and $\tan(x)$ :
Rearrange the expression for $\tan(x)$ :
$\sin(x) = \tan(x) \cdot \cos(x)$
Since $\csc(x) = \frac{1}{\sin(x)}$ , substitute the expression for $\sin(x)$ :
$\csc(x) = \frac{1}{\tan(x) \cdot \cos(x)}$
Thus, $\csc(x) = \frac{1}{\tan(x) \cdot \cos(x)}$ .
The expressions are: a. $\cos(x) = \frac{\cot(x)}{\csc(x)}$ b. $\csc(x) = \frac{1}{\tan(x) \cdot \cos(x)}$

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Question1. a. Express $\cos (x)$ in terms of $\cot (x)$ and $\csc (x)$ . b. Express $\csc (x)$ in terms of $\cos (x)$ and $\tan (x)$ .

Studdy Solution

STEP 1

1. We are working with trigonometric identities and relationships.
2. We will use basic trigonometric identities to express one function in terms of others.
3. We assume $x$ is an angle where all trigonometric functions are defined.

STEP 2

1. Express $\cos(x)$ in terms of $\cot(x)$ and $\csc(x)$ .
2. Express $\csc(x)$ in terms of $\cos(x)$ and $\tan(x)$ .

STEP 3

Start with the identity for $\cot(x)$ and $\csc(x)$ :
$\cot(x) = \frac{\cos(x)}{\sin(x)}$
$\csc(x) = \frac{1}{\sin(x)}$

STEP 4

STEP 5

Start with the identity for $\tan(x)$ and $\cos(x)$ :
$\tan(x) = \frac{\sin(x)}{\cos(x)}$

STEP 6

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Question1. a. Express cos⁡(x)\cos (x)cos(x) in terms of cot⁡(x)\cot (x)cot(x) and csc⁡(x)\csc (x)csc(x). b. Express csc⁡(x)\csc (x)csc(x) in terms of cos⁡(x)\cos (x)cos(x) and tan⁡(x)\tan (x)tan(x).

Studdy Solution

STEP 1

1. We are working with trigonometric identities and relationships.2. We will use basic trigonometric identities to express one function in terms of others.3. We assume x x x is an angle where all trigonometric functions are defined.

STEP 2

1. Express cos⁡(x)\cos(x)cos(x) in terms of cot⁡(x)\cot(x)cot(x) and csc⁡(x)\csc(x)csc(x).2. Express csc⁡(x)\csc(x)csc(x) in terms of cos⁡(x)\cos(x)cos(x) and tan⁡(x)\tan(x)tan(x).

STEP 3

Start with the identity for cot⁡(x)\cot(x)cot(x) and csc⁡(x)\csc(x)csc(x):cot⁡(x)=cos⁡(x)sin⁡(x)\cot(x) = \frac{\cos(x)}{\sin(x)}cot(x)=sin(x)cos(x)​csc⁡(x)=1sin⁡(x)\csc(x) = \frac{1}{\sin(x)}csc(x)=sin(x)1​

STEP 4

STEP 5

Start with the identity for tan⁡(x)\tan(x)tan(x) and cos⁡(x)\cos(x)cos(x):tan⁡(x)=sin⁡(x)cos⁡(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)​

STEP 6

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Question1. a. Express $\cos (x)$ in terms of $\cot (x)$ and $\csc (x)$ . b. Express $\csc (x)$ in terms of $\cos (x)$ and $\tan (x)$ .

1. We are working with trigonometric identities and relationships.
2. We will use basic trigonometric identities to express one function in terms of others.
3. We assume $x$ is an angle where all trigonometric functions are defined.

1. Express $\cos(x)$ in terms of $\cot(x)$ and $\csc(x)$ .
2. Express $\csc(x)$ in terms of $\cos(x)$ and $\tan(x)$ .

Start with the identity for $\cot(x)$ and $\csc(x)$ :
$\cot(x) = \frac{\cos(x)}{\sin(x)}$
$\csc(x) = \frac{1}{\sin(x)}$

Start with the identity for $\tan(x)$ and $\cos(x)$ :
$\tan(x) = \frac{\sin(x)}{\cos(x)}$