What is 1 / cos x = sec x? | Simple Explanation for Class 10

S6-SA2-0409

What is the Identity 1 / cos x = sec x?

Grade Level:

Class 10

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

Definition

What is it?

The identity 1 / cos x = sec x tells us that the secant of an angle (sec x) is simply the reciprocal of the cosine of that same angle (cos x). This means if you know the value of cos x, you can easily find sec x by dividing 1 by that value. It's one of the fundamental relationships in trigonometry.

Simple Example

Quick Example

Imagine your friend scores 50 runs in a cricket match. If someone asks for the 'reciprocal' of his score, it would be 1/50. Similarly, if cos x is like a score, then sec x is its reciprocal, 1 / cos x. So, if cos x = 1/2, then sec x would be 1 / (1/2) = 2.

Worked Example

Step-by-Step

Let's find the value of sec x if cos x = 0.8.
---Step 1: Recall the identity: sec x = 1 / cos x.
---Step 2: Substitute the given value of cos x into the identity. So, sec x = 1 / 0.8.
---Step 3: Perform the division. 1 divided by 0.8 is 1.25.
---Step 4: State the answer clearly. Therefore, if cos x = 0.8, then sec x = 1.25.

Why It Matters

Understanding this identity is crucial for solving complex problems in engineering and physics, like calculating forces or wave patterns. Engineers use it to design stable bridges, while scientists in space technology use it to track satellite orbits. It's a building block for careers in AI/ML, robotics, and even medicine for analyzing biological signals.

Common Mistakes

MISTAKE: Thinking sec x = cos x | CORRECTION: Remember sec x is the RECIPROCAL of cos x, meaning sec x = 1 / cos x, not equal to it.

MISTAKE: Calculating sec x as 1 - cos x | CORRECTION: The operation is division, not subtraction. It's 1 DIVIDED BY cos x.

MISTAKE: Using the angle value in the calculation (e.g., sec 30 = 1 / 30) | CORRECTION: You need to find the COSINE of the angle first (e.g., cos 30 degrees), then take its reciprocal. So, sec 30 = 1 / (cos 30 degrees).

Practice Questions

Try It Yourself

QUESTION: If cos x = 0.5, what is sec x? | ANSWER: 2

QUESTION: Find the value of sec x when cos x = 4/5. | ANSWER: 5/4 or 1.25

QUESTION: If the value of cos x is 0.25, and then x doubles (so now we need cos 2x), will sec 2x be double of sec x? (Assume cos 2x = -0.6 for this specific problem) | ANSWER: No. If cos x = 0.25, sec x = 4. If cos 2x = -0.6, then sec 2x = 1 / -0.6 = -1.67 (approx). They are not simply double.

MCQ

Quick Quiz

Which of the following is equivalent to 1 / cos x?

sin x

tan x

sec x

cot x

The Correct Answer Is:

The identity states that sec x is the reciprocal of cos x. Therefore, 1 / cos x is equal to sec x. The other options are different trigonometric ratios.

Real World Connection

In the Real World

This identity is used in computer graphics and animation to calculate light reflection angles on surfaces. For example, when you play a video game on your mobile, the way light shines on a car or a building is often calculated using trigonometric functions, including secant, to make the graphics look realistic.

Key Vocabulary

Key Terms

IDENTITY: A mathematical equation that is true for all possible values of its variables. | RECIPROCAL: The number by which another number must be multiplied to produce 1 (e.g., reciprocal of 2 is 1/2). | COSINE (cos x): One of the main trigonometric ratios, relating an angle to the ratio of the adjacent side to the hypotenuse in a right-angled triangle. | SECANT (sec x): One of the main trigonometric ratios, which is the reciprocal of the cosine.

What's Next

What to Learn Next

Great job understanding sec x! Next, explore the other reciprocal identities like 1 / sin x = cosec x and 1 / tan x = cot x. These will complete your understanding of how all trigonometric ratios are connected, which is super helpful for more advanced trigonometry.