Trigonometric Functions

To determine the angle of elevation of the sun, we can use trigonometric principles. In this scenario, we have a right triangle formed by the height of the pool, its shadow, and the hypotenuse (line from the top of the pool to the end of the shadow). The shadow length given is 10√3 meters, and the pool's height is 30 meters.

To find the angle of elevation θ (the angle between the shadow and the line stretching from the pool top to shadow tip), we use the tangent function:

tan(θ) = Opposite/Adjacent

tan(θ) = Height of pool / Length of shadow = 30 / (10√3)

Simplify the expression:

tan(θ) = 30 / (10√3) = 3/√3

Further simplification yields:

tan(θ) = √3

From trigonometric tables, we know that:

tan(60°) = √3

Therefore, the angle of elevation θ is 60°.

To find the angle of elevation of the sun, we need to understand the relationship between the height of the tower, the length of the shadow, and the angle of elevation. This can be done using trigonometry, specifically the tangent function, which relates the opposite side (height of the tower) to the adjacent side (length of the shadow) in a right-angled triangle.

The formula is:

tan(θ) = Opposite/Adjacent

Here,

• Opposite = 45 cm (height of the tower)
• Adjacent = 12 cm (length of the shadow)

Substituting these values into the formula gives:

tan(θ) = 45/12

Simplifying the fraction:

tan(θ) = 3.75

Now, to find θ, take the arctangent (inverse tangent) of 3.75:

θ = arctan(3.75)

Using a calculator,

θ ≈ 75°

Therefore, the angle of elevation of the sun is 75°.

To evaluate the expression , we'll use trigonometric identities and simplification techniques.

1. Given Expression:
We need to find the value of:

2. Using Trigonometric Identity:
Multiply numerator and denominator by :

Let

3. Applying Double Angle Formula:
Using :

4. Repeat the Process:
Multiply numerator and denominator by 2:

5. Continue Simplification:
Multiply numerator and denominator by 2 again:

6. Final Simplification:
Using :

7. Using Complementary Angle:
Recognize that when combined with
Thus

8. Calculating the Value:

Therefore:

Final Answer:
The value of is .

We need to evaluate the expression .

1. Simplify Using Trigonometric Identities:
Notice the angles: , and we can explore relationships with and . Use the co-function identity for cosine:


The expression becomes:

2. Examine the Angles:
Notice that , and , suggesting a possible relationship. Let’s try to use sum-to-product identities for the sine terms:


Use the identity , where , :

,
Thus:


Since , we get:

3. Substitute Back:
The original expression is now:

4. Verify the Result:
The cancellation suggests the expression simplifies to zero. To confirm, we could compute numerically, but the algebraic simplification is consistent, and the angles’ relationships support this outcome.

Final Answer:
The value of is .