Trigonometric functions (sin, cos, tan) in NumPy

Introduction

Trigonometric functions help us find relationships between angles and sides in triangles. They are useful for analyzing waves, rotations, and cycles in data.

To calculate the height of a tree using its shadow and angle of elevation.

To analyze periodic data like sound waves or seasonal sales patterns.

To rotate points in a 2D plane for graphics or simulations.

To model circular motion or oscillations in physics or engineering.

Syntax

NumPy

import numpy as np

np.sin(angle_in_radians)
np.cos(angle_in_radians)
np.tan(angle_in_radians)

Input angles must be in radians, not degrees.

Use np.radians(degrees) to convert degrees to radians.

Examples

Calculate sin, cos, tan for 90 degrees (π/2 radians).

NumPy

import numpy as np

angle = np.pi / 2  # 90 degrees in radians
sin_val = np.sin(angle)
cos_val = np.cos(angle)
tan_val = np.tan(angle)
print(sin_val, cos_val, tan_val)

Convert 45 degrees to radians and find its sine.

NumPy

import numpy as np

degrees = 45
radians = np.radians(degrees)
sin_45 = np.sin(radians)
print(sin_45)

Calculate sine for multiple angles at once using an array.

NumPy

import numpy as np

angles = np.array([0, np.pi/4, np.pi/2])
sin_values = np.sin(angles)
print(sin_values)

Sample Program

This program converts a list of angles from degrees to radians, then calculates and prints their sine, cosine, and tangent values in a clear table.

NumPy

import numpy as np

# Angles in degrees
angles_degrees = np.array([0, 30, 45, 60, 90])

# Convert degrees to radians
angles_radians = np.radians(angles_degrees)

# Calculate sine, cosine, and tangent
sin_values = np.sin(angles_radians)
cos_values = np.cos(angles_radians)
tan_values = np.tan(angles_radians)

# Print results in a table format
print("Angle (deg) |   sin   |   cos   |   tan")
print("---------------------------------------")
for deg, s, c, t in zip(angles_degrees, sin_values, cos_values, tan_values):
    print(f"{deg:10} | {s:7.3f} | {c:7.3f} | {t:7.3f}")

Important Notes

Tangent of 90 degrees (π/2 radians) is very large because it approaches infinity.

Always convert degrees to radians before using numpy trigonometric functions.

Use numpy arrays to calculate trig functions on many angles efficiently.

Summary

Trigonometric functions relate angles to ratios of sides in triangles.

Use numpy's sin, cos, and tan functions with angles in radians.

Convert degrees to radians with np.radians() before calculations.

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()