PIL import Image

Let's open and display the image to understand its content.

image_path = '/mnt/data/1733573333098bxak9vom.jpg' img = Image.open(image_path) img.show()It seems that I need to interpret the image content in order to assist you with the mathematical calculations. I will extract and analyze the image to identify the relevant expressions. Let me process it for you. pytesseract

Using pytesseract to extract text from the image.

extracted_text = pytesseract.image_to_string(img)

extracted_textThe extracted expression from the image is:

$\sin^{-1}(8^\circ) + 4\cos(5^\circ)$

Now, to compute this approximation using the total differential, we need to apply the following steps:

1. Total Differential Approach: We will approximate the change in the expression by calculating the partial derivatives with respect to each variable and multiplying by the small change in each variable.

2. First, we simplify: We need to evaluate both terms in the expression (inverse sine and cosine) numerically. This involves using approximations for small angles and applying the known values for sine and cosine.

Let me compute these values and provide an approximation for the expression. math

Define the angles in degrees and convert them to radians for the trigonometric functions

angle_asin = 8 # degrees angle_cos = 5 # degrees

Convert to radians

angle_asin_rad = math.radians(angle_asin) angle_cos_rad = math.radians(angle_cos)

Compute the values of the inverse sine and cosine

asin_value = math.asin(math.sin(angle_asin_rad)) # asin(8°) in radians cos_value = math.cos(angle_cos_rad) # cos(5°) in radians

Now compute the full expression

expression_value = asin_value + 4 * cos_value

Return the calculated values

asin_value_deg = math.degrees(asin_value) # Convert back to degrees for clarity asin_value_deg, cos_value, expression_valueThe calculations are as follows:

1. The value of $\sin^{-1}(8^\circ)$ is approximately $8^\circ$ (since the sine of $8^\circ$ is small, the inverse sine is nearly the same value).
2. The value of $\cos(5^\circ)$ is approximately $0.9962$.

Thus, the total expression:

$\sin^{-1}(8^\circ) + 4\cos(5^\circ) \approx 8^\circ + 4(0.9962) = 8^\circ + 3.9848 = 4.1244 \text{ (in radians)}.$

If you need further details or clarifications on this, feel free to ask! Here are some related questions to explore:

1. How to compute the inverse sine (arc sine) of other values?
2. What is the importance of small-angle approximations in trigonometry?
3. How do we use differential approximations in other types of calculus problems?
4. How would the approximation change with a different angle (e.g., 10°)?
5. What are the advantages of using radians instead of degrees in calculus?

Tip: For small angles, using the approximation $\sin(\theta) \approx \theta$ (in radians) can make calculations simpler, especially for small values like 8°.