What is Trigonometric Ratios for (90° + A)?

S6-SA2-0414

Grade Level:

Class 10

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

Definition

What is it?

Trigonometric Ratios for (90° + A) tell us how the sine, cosine, and tangent of an angle (90° + A) relate to the trigonometric ratios of the angle 'A' itself. These relationships help us find the values of trigonometric functions for angles greater than 90 degrees by using values from 0 to 90 degrees.

Simple Example

Quick Example

Imagine you are watching a cricket match and want to calculate the angle a ball makes with the ground after bouncing. If the initial angle 'A' was 30 degrees, then (90° + A) would be 120 degrees. Using these ratios, you can find the sin, cos, or tan of 120 degrees by knowing the sin, cos, or tan of 30 degrees.

Worked Example

Step-by-Step

Let's find the value of sin(120°).
---Step 1: Recognize that 120° can be written as (90° + 30°). So, A = 30°.
---Step 2: Recall the formula for sin(90° + A), which is cos(A).
---Step 3: Substitute A = 30° into the formula: sin(90° + 30°) = cos(30°).
---Step 4: We know that cos(30°) = sqrt(3)/2.
---Step 5: Therefore, sin(120°) = sqrt(3)/2.
Answer: sin(120°) = sqrt(3)/2

Why It Matters

Understanding these ratios is crucial for fields like Physics and Engineering, where you calculate forces and trajectories. For example, Space Technology uses these concepts to guide rockets, and AI/ML models can use them for image processing. They are also vital for careers in architecture and navigation.

Common Mistakes

MISTAKE: Confusing the sign of the ratio (positive or negative) in different quadrants. For example, thinking sin(90° + A) is -cos(A). | CORRECTION: Remember that in the second quadrant (where 90° + A lies), sine is positive, while cosine and tangent are negative.

MISTAKE: Incorrectly switching the trigonometric function. For example, writing cos(90° + A) = sin(A). | CORRECTION: The rule for 90° + A is that sine changes to cosine, cosine changes to sine, and tangent changes to cotangent. So, cos(90° + A) = -sin(A).

MISTAKE: Not knowing the standard values for angles like 0°, 30°, 45°, 60°, 90°. | CORRECTION: Memorize the trigonometric values for these basic angles. They are the building blocks for solving more complex problems.

Practice Questions

Try It Yourself

QUESTION: What is the value of cos(150°)? | ANSWER: -sqrt(3)/2

QUESTION: If tan(A) = 1, what is the value of tan(90° + A)? | ANSWER: -1

QUESTION: If sin(A) = 1/2, find the value of [sin(90° + A)] * [cos(90° + A)]. | ANSWER: -sqrt(3)/4

MCQ

Quick Quiz

Which of the following is correct?

sin(90° + A) = -cos(A)

cos(90° + A) = sin(A)

tan(90° + A) = -cot(A)

cot(90° + A) = tan(A)

The Correct Answer Is:

In the second quadrant (90° + A), tangent is negative, and tan changes to cot. So, tan(90° + A) = -cot(A) is correct. Options A, B, and D have incorrect signs or function changes.

Real World Connection

In the Real World

When ISRO launches satellites, engineers constantly calculate the angle of elevation and trajectory. These calculations often involve angles greater than 90 degrees. Knowing how to use trigonometric ratios for (90° + A) helps them predict where the satellite will be and adjust its path precisely.

Key Vocabulary

Key Terms

Trigonometric Ratios: relationships between angles and sides of a right-angled triangle | Quadrant: one of four sections of a coordinate plane | Angle A: an acute angle used as a reference | Sine: opposite side / hypotenuse | Cosine: adjacent side / hypotenuse

What's Next

What to Learn Next

Next, you should explore Trigonometric Ratios for (180° - A) and (180° + A). These concepts build directly on what you've learned here, helping you understand how trigonometric values behave across all four quadrants.