# Precalculus : Fundamental Trigonometric Identities

## Example Questions

### Example Question #31 : Fundamental Trigonometric Identities

Let , , and  be real numbers. Given that:

What is the value of  in function of ?

Explanation:

We note first, using trigonometric identities that:

This gives:

Since,

We have :

### Example Question #31 : Fundamental Trigonometric Identities

Using the fact that,

.

What is the result of the following sum:

Explanation:

We can write the above sum as :

From the given fact, we have :

and we have : .

This gives :

### Example Question #33 : Fundamental Trigonometric Identities

Compute  in function of .

Explanation:

Using trigonometric identities we have :

and we know that:

This gives us :

Hence:

### Example Question #34 : Fundamental Trigonometric Identities

Given that :

Let,

What is  in function of ?

Explanation:

We will use the given formula :

We have in this case:

Since we know that :

This gives :

### Example Question #35 : Fundamental Trigonometric Identities

Using the fact that  , what is the result of the following sum:



Explanation:

We can write the above sum as :

From the given fact, we have :

This gives us :

Therefore we have:

### Example Question #36 : Fundamental Trigonometric Identities

Let be real numbers. If  and

What is the value of in function of   ?

Explanation:

Using trigonometric identities we know that :

This gives :

We also know that

This gives :

### Example Question #37 : Fundamental Trigonometric Identities

Given that :

and,

Compute :

in function of .

Explanation:

We have using the given result:

This gives us:

Hence :

### Example Question #38 : Fundamental Trigonometric Identities

Let  be an integer and  a real number. Compute  as a function of .

Explanation:

Using trigonometric identities we have :

We know that :

and

This gives :

### Example Question #39 : Fundamental Trigonometric Identities

Compute .

Explanation:

Using trigonometric identities we know that:

Letting  and  in the above expression we have:

We also know that:

and .

This gives:

### Example Question #40 : Fundamental Trigonometric Identities

Given that:

, what is the value of  in function of ?

Explanation:

We know by definition that:

We also have by trigonometric identities:

Thus :

Now we have:

This gives us: