Why Trigonometric functions (sin, cos, tan) in NumPy? - Purpose & Use Cases

The Big Idea

What if you could instantly calculate heights and distances from angles without any manual math errors?

The Scenario

Imagine you want to calculate the height of a tree using angles you measured with a protractor. Doing this by hand means using a calculator for each angle, writing down results, and repeating for many trees or measurements.

The Problem

Manually calculating sine, cosine, or tangent for many angles is slow and easy to mess up. You might type wrong numbers, lose track of results, or spend hours repeating the same steps.

The Solution

Using trigonometric functions in numpy lets you calculate sin, cos, and tan for many angles at once, quickly and without mistakes. It automates the math so you can focus on understanding the results.

Before vs After

✗ Before

angle = 30
sin_val = 0.5  # looked up or calculated manually

✓ After

import numpy as np
angle = 30
sin_val = np.sin(np.radians(angle))

What It Enables

It enables fast, accurate calculations of angles and distances for many data points, unlocking powerful analysis in science, engineering, and everyday problems.

Real Life Example

Surveyors use trigonometric functions to find distances and heights of objects they cannot measure directly, like tall buildings or mountains, by measuring angles from a distance.

Key Takeaways

Manual angle calculations are slow and error-prone.

Numpy's trig functions automate and speed up these calculations.

This helps solve real-world problems involving angles and distances efficiently.

Practice

1. What does the numpy function np.sin() calculate when given an angle in radians?

easy

A. The angle converted from radians to degrees.

B. The cosine of the angle, which is the ratio of the adjacent side to the hypotenuse.

C. The tangent of the angle, which is the ratio of the opposite side to the adjacent side.

D. The sine of the angle, which is the ratio of the opposite side to the hypotenuse in a right triangle.

Solution

1. Step 1: Understand the function purpose

   The function np.sin() calculates the sine value of an angle given in radians.

2. Step 2: Recall sine definition in triangles

   Sine of an angle is the ratio of the length of the opposite side to the hypotenuse in a right triangle.

3. Final Answer:

   The sine of the angle, which is the ratio of the opposite side to the hypotenuse in a right triangle. -> Option D

4. Quick Check:

   np.sin() gives sine ratio [OK]

Hint: Remember sin = opposite/hypotenuse in triangles [OK]

Common Mistakes:

• Confusing sine with cosine or tangent
• Thinking np.sin() converts radians to degrees
• Using degrees directly without conversion

2. Which of the following is the correct way to calculate the cosine of 60 degrees using numpy?

easy

A. np.cos(np.radians(60))

B. np.cos(np.sin(60))

C. np.cos(np.degrees(60))

D. np.cos(60)

Solution

1. Step 1: Recognize angle units for numpy trig functions

   numpy trigonometric functions expect angles in radians, not degrees.

2. Step 2: Convert degrees to radians before using np.cos()

   Use np.radians(60) to convert 60 degrees to radians, then apply np.cos().

3. Final Answer:

   np.cos(np.radians(60)) -> Option A

4. Quick Check:

   Convert degrees to radians before trig functions [OK]

Hint: Always convert degrees to radians with np.radians() [OK]

Common Mistakes:

• Passing degrees directly to np.cos()
• Using np.degrees() instead of np.radians()
• Passing a trig function inside np.cos()

3. What is the output of the following code?

import numpy as np
angles = np.array([0, 90, 180])
radians = np.radians(angles)
sin_values = np.sin(radians)
print(np.round(sin_values, 2))

medium

A. [0.00 0.00 0.00]

B. [1.00 0.00 -1.00]

C. [0.00 1.00 0.00]

D. [0.00 -1.00 0.00]

Solution

1. Step 1: Convert angles to radians

   Angles 0, 90, 180 degrees are converted to radians: 0, π/2, π.

2. Step 2: Calculate sine values and round

   sin(0) = 0, sin(π/2) = 1, sin(π) = 0. Rounded to 2 decimals: [0.00, 1.00, 0.00].

3. Final Answer:

   [0.00 1.00 0.00] -> Option C

4. Quick Check:

   sin(0, 90, 180) = [0, 1, 0] [OK]

Hint: Recall sin(0)=0, sin(90°)=1, sin(180°)=0 [OK]

Common Mistakes:

• Not converting degrees to radians
• Confusing sine values at 90 and 180 degrees
• Forgetting to round output

4. The following code is intended to calculate the tangent of 45 degrees but gives an incorrect result. What is the error?

import numpy as np
tan_45 = np.tan(45)
print(tan_45)

medium

A. np.tan() cannot calculate tangent for 45 degrees.

B. The angle 45 should be converted to radians before using np.tan().

C. The code should use np.tan(np.degrees(45)) instead.

D. The print statement is missing parentheses.

Solution

1. Step 1: Identify input units for np.tan()

   numpy trigonometric functions require angles in radians, not degrees.

2. Step 2: Correct the input by converting degrees to radians

   Use np.radians(45) to convert 45 degrees before passing to np.tan().

3. Final Answer:

   The angle 45 should be converted to radians before using np.tan(). -> Option B

4. Quick Check:

   Convert degrees to radians before np.tan() [OK]

Hint: Always convert degrees to radians before trig functions [OK]

Common Mistakes:

• Passing degrees directly to np.tan()
• Using np.degrees() instead of np.radians()
• Assuming np.tan() works with degrees

5. You have an array of angles in degrees: angles = np.array([30, 45, 60]). You want to create a new array that contains the tangent values of these angles but only include values where the tangent is less than 2. Which code correctly does this?

hard

A. tan_vals = np.tan(np.radians(angles)); filtered = tan_vals[tan_vals < 2]

B. tan_vals = np.tan(angles); filtered = tan_vals[tan_vals < 2]

C. tan_vals = np.tan(np.degrees(angles)); filtered = tan_vals[tan_vals < 2]

D. tan_vals = np.tan(np.radians(angles)); filtered = tan_vals[tan_vals > 2]

Solution

1. Step 1: Convert degrees to radians for tangent calculation

   Use np.radians(angles) to convert the array of degrees to radians before applying np.tan().

2. Step 2: Filter tangent values less than 2

   Use boolean indexing tan_vals[tan_vals < 2] to select values less than 2.

3. Final Answer:

   tan_vals = np.tan(np.radians(angles)); filtered = tan_vals[tan_vals < 2] -> Option A

4. Quick Check:

   Convert degrees, then filter tan values less than 2 [OK]

Hint: Convert degrees first, then filter with boolean indexing [OK]

Common Mistakes:

• Not converting degrees to radians
• Filtering with wrong comparison operator
• Using np.degrees() instead of np.radians()