What is sin x / cos x = tan x? | Simple Explanation for Class 10

S6-SA2-0407

What is the Identity sin x / cos x = tan x?

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The identity sin x / cos x = tan x is a fundamental relationship in trigonometry that connects the sine, cosine, and tangent ratios for any angle x. It means that if you divide the sine of an angle by the cosine of the same angle, you will always get the tangent of that angle. This identity is true for all angles where cos x is not zero.

Simple Example

Quick Example

Imagine you're watching a cricket match and the commentator gives you two numbers: the 'run rate' (like sin x) and the 'overs bowled' (like cos x). If you divide the run rate by the overs bowled, you might get a 'strike rate' (like tan x) that tells you how fast the scoring is. Similarly, sin x divided by cos x always gives tan x for a given angle.

Worked Example

Step-by-Step

Let's verify the identity sin x / cos x = tan x for a specific angle, say x = 30 degrees.

Step 1: Find the value of sin 30 degrees. From the trigonometry table, sin 30 degrees = 1/2.

---Step 2: Find the value of cos 30 degrees. From the trigonometry table, cos 30 degrees = sqrt(3)/2.

---Step 3: Divide sin 30 degrees by cos 30 degrees. (1/2) / (sqrt(3)/2).

---Step 4: Simplify the division. (1/2) * (2/sqrt(3)) = 1/sqrt(3).

---Step 5: Find the value of tan 30 degrees. From the trigonometry table, tan 30 degrees = 1/sqrt(3).

---Step 6: Compare the results. Since (sin 30 degrees / cos 30 degrees) = 1/sqrt(3) and tan 30 degrees = 1/sqrt(3), the identity holds true for x = 30 degrees.

Answer: The identity is verified.

Why It Matters

This identity is super important in many fields, from designing rockets to creating video games. Engineers use it to calculate forces and angles in structures, while physicists apply it to understand wave patterns and light. Knowing this helps you build strong foundations for careers in engineering, space technology, and even AI/ML where understanding spatial relationships is key.

Common Mistakes

MISTAKE: Students often confuse the order and write cos x / sin x = tan x. | CORRECTION: Remember it's always sin x divided by cos x to get tan x. Think 'S' before 'C' alphabetically.

MISTAKE: Assuming the identity works for different angles, like sin A / cos B = tan A. | CORRECTION: The angle 'x' must be the same for sine, cosine, and tangent in this identity. It's sin x / cos x = tan x, not sin A / cos B.

MISTAKE: Forgetting that tan x is undefined when cos x is zero (e.g., at 90 degrees or 270 degrees). | CORRECTION: Always remember that division by zero is not allowed. So, this identity holds true only when cos x is not equal to zero.

Practice Questions

Try It Yourself

QUESTION: If sin A = 3/5 and cos A = 4/5, what is tan A? | ANSWER: 3/4

QUESTION: For an angle x where tan x = 1, and cos x = 1/sqrt(2), what is sin x? | ANSWER: 1/sqrt(2)

QUESTION: If sin theta = 0.8 and cos theta = 0.6, calculate tan theta. Then, if you know a right-angled triangle has an opposite side of 4 units and an adjacent side of 3 units for angle theta, does your calculated tan theta match? | ANSWER: tan theta = 0.8 / 0.6 = 4/3. Yes, it matches because tan theta (from triangle) = Opposite/Adjacent = 4/3.

MCQ

Quick Quiz

Which of the following is equivalent to tan A?

cos A / sin A

1 / cot A

sin A * cos A

1 / sin A

The Correct Answer Is:

Option B is correct because tan A is the reciprocal of cot A, so tan A = 1/cot A. While sin A / cos A is also equal to tan A, it's not an option here. Options A, C, and D are incorrect mathematical relationships.

Real World Connection

In the Real World

This identity helps engineers design ramps and roads. When building a flyover in a city like Bangalore, civil engineers use trigonometry to calculate the slope (which relates to tan x) required for a smooth ascent, considering the vertical rise (sin x) and horizontal distance covered (cos x). This ensures vehicles can climb safely and efficiently.

Key Vocabulary

Key Terms

IDENTITY: A mathematical equation that is true for all possible values of its variables | SINE: The ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle | COSINE: The ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle | TANGENT: The ratio of the length of the opposite side to the length of the adjacent side in a right-angled triangle | TRIGONOMETRY: The branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles.

What's Next

What to Learn Next

Great job understanding this basic identity! Next, you should explore other important trigonometric identities like sin^2 x + cos^2 x = 1. These identities are like building blocks that will help you solve more complex problems in geometry and physics.