Trigonometric functions (sin, cos, tan) in NumPy - Practice Problems & Coding Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of numpy sine function on array

What is the output of this code snippet using numpy's sine function on an array of angles in radians?

NumPy

import numpy as np
angles = np.array([0, np.pi/2, np.pi])
result = np.sin(angles)
print(result)

A [1. 0. 1. ]

B [0. 0.70710678 1. ]

C [0. 1. 0. ]

D [0. 1. -1. ]

💡 Hint

Recall sine values at 0, 90, and 180 degrees (in radians: 0, π/2, π).

The sine of 0 is 0, sine of π/2 (90 degrees) is 1, and sine of π (180 degrees) is 0.

❓ data_output

intermediate

Number of elements in cosine output array

After running this code, how many elements does the resulting cosine array contain?

NumPy

import numpy as np
angles = np.linspace(0, 2*np.pi, 50)
cos_values = np.cos(angles)
print(len(cos_values))

A 50

B 49

C 51

D 52

💡 Hint

np.linspace(start, stop, num) creates 'num' evenly spaced points.

np.linspace with num=50 creates exactly 50 points, so the cosine array has 50 elements.

🔧 Debug

advanced

Identify the error in tangent calculation

What error will this code raise when calculating tangent values for these angles?

NumPy

import numpy as np
angles = np.array([0, np.pi/2, np.pi])
tan_values = np.tan(angles)
print(tan_values)

A ValueError: invalid value in tan

B No error, outputs finite values

C TypeError: unsupported operand type(s) for tan

D RuntimeWarning: divide by zero encountered in tan

💡 Hint

Consider the tangent of π/2 (90 degrees).

Tangent of π/2 is undefined (infinite), so numpy raises a RuntimeWarning for divide by zero.

🧠 Conceptual

advanced

Understanding output range of cosine function

Which of the following statements about the output range of numpy's cosine function is correct?

A The output values are always between 0 and 1 inclusive.

B The output values are always between -1 and 1 inclusive.

C The output values can be any real number depending on input.

D The output values are always positive.

💡 Hint

Recall the range of the cosine function in trigonometry.

Cosine values range from -1 to 1 for all real inputs.

🚀 Application

expert

Calculate and plot sine and cosine waves

Which option correctly calculates sine and cosine values for 100 points between 0 and 2π and plots both on the same graph?

A

import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(0, 2*np.pi, 100)
plt.plot(x, np.sin(x), label='sin')
plt.plot(x, np.cos(x), label='cos')
plt.legend()
plt.show()

B

import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(0, 2*np.pi, 100)
plt.plot(np.sin(x), x, label='sin')
plt.plot(np.cos(x), x, label='cos')
plt.legend()
plt.show()

C

import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(0, 2*np.pi, 100)
plt.plot(x, np.sin(x))
plt.plot(x, np.cos(x))
plt.show()

D

import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(0, 2*np.pi, 100)
plt.plot(x, np.tan(x), label='tan')
plt.plot(x, np.sin(x), label='sin')
plt.legend()
plt.show()

💡 Hint

Check if both sine and cosine are plotted against the same x values and if labels are included.

Option A correctly plots sine and cosine against x with labels and legend. Option A swaps axes, A plots tangent which can be problematic, D misses labels and legend.

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()