What is Finding Unknown Sides using Trigonometric Ratios?

S6-SA2-0075

Grade Level:

Class 10

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

Definition

What is it?

Finding unknown sides using trigonometric ratios means using the relationships between the angles and sides of a right-angled triangle to calculate a side whose length we don't know. We use special ratios like sine (sin), cosine (cos), and tangent (tan) for this.

Simple Example

Quick Example

Imagine you're flying a kite, and its string is 50 meters long. You estimate the angle the string makes with the ground is 30 degrees. How high is the kite? Trigonometric ratios help you find this unknown height without actually climbing up there!

Worked Example

Step-by-Step

PROBLEM: A ladder is leaning against a wall. The ladder is 10 meters long and makes an angle of 60 degrees with the ground. How high up the wall does the ladder reach?

Step 1: Draw a right-angled triangle. The ladder is the hypotenuse (10 m), the wall is the 'opposite' side to the 60-degree angle, and the ground is the 'adjacent' side.
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Step 2: Identify what you know and what you need to find. We know the hypotenuse (10 m) and the angle (60 degrees). We need to find the height the ladder reaches on the wall, which is the 'opposite' side.
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Step 3: Choose the correct trigonometric ratio. The ratio that relates the 'opposite' side and the 'hypotenuse' is sine (sin). So, sin(angle) = Opposite / Hypotenuse.
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Step 4: Substitute the known values into the formula. sin(60 degrees) = Height / 10.
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Step 5: Look up the value of sin(60 degrees). sin(60 degrees) is approximately 0.866.
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Step 6: Solve for the unknown side (Height). 0.866 = Height / 10. So, Height = 0.866 * 10.
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Step 7: Calculate the final answer. Height = 8.66 meters.

ANSWER: The ladder reaches 8.66 meters high on the wall.

Why It Matters

This concept is super important for many fields! Engineers use it to design buildings and bridges, ensuring they are stable. In space technology, ISRO scientists use it to calculate distances to satellites. Even in medicine, doctors might use similar principles for imaging or understanding body mechanics.

Common Mistakes

MISTAKE: Confusing which side is 'opposite' or 'adjacent' to the given angle. | CORRECTION: Always identify the angle you are working with first. The side directly across from it is the 'opposite' side. The side next to it that is NOT the hypotenuse is the 'adjacent' side.

MISTAKE: Using the wrong trigonometric ratio (e.g., using cos instead of sin). | CORRECTION: Remember SOH CAH TOA! SOH (Sin = Opposite/Hypotenuse), CAH (Cos = Adjacent/Hypotenuse), TOA (Tan = Opposite/Adjacent). Choose the ratio that includes the sides you know and the side you need to find.

MISTAKE: Forgetting to put the calculator in 'degree' mode. | CORRECTION: Most scientific calculators have modes for degrees and radians. Always ensure your calculator is set to 'DEG' (degrees) when solving these problems, especially for Class 10.

Practice Questions

Try It Yourself

QUESTION: A tree casts a shadow 15 meters long when the sun is at an angle of elevation of 45 degrees. How tall is the tree? | ANSWER: 15 meters

QUESTION: An airplane takes off at an angle of 15 degrees. If it has flown 2000 meters along its path, what is its current altitude (height above the ground)? (Use sin 15 degrees = 0.2588) | ANSWER: 517.6 meters

QUESTION: A 25-meter rope is tied from the top of a pole to a point on the ground. If the rope makes an angle of 30 degrees with the ground, how far is the base of the pole from the point where the rope is tied? (Use cos 30 degrees = 0.866) | ANSWER: 21.65 meters

MCQ

Quick Quiz

In a right-angled triangle, if the angle is 30 degrees and the side opposite to it is 5 cm, what is the length of the hypotenuse?

5 / cos(30 degrees)

5 * sin(30 degrees)

5 / sin(30 degrees)

5 * tan(30 degrees)

The Correct Answer Is:

We know the opposite side and need the hypotenuse. The ratio relating these is sin(angle) = Opposite/Hypotenuse. So, sin(30) = 5/Hypotenuse, which means Hypotenuse = 5/sin(30).

Real World Connection

In the Real World

Surveyors use trigonometry to measure land boundaries and heights of mountains without physically climbing them. For example, a surveyor in Rajasthan might use a theodolite (a special instrument) to measure angles and then apply trigonometric ratios to find the width of a river or the height of a distant monument like Qutub Minar from a safe distance.

Key Vocabulary

Key Terms

RIGHT-ANGLED TRIANGLE: A triangle with one angle exactly 90 degrees. | HYPOTENUSE: The longest side of a right-angled triangle, opposite the 90-degree angle. | SINE (sin): Ratio of the length of the opposite side to the length of the hypotenuse. | COSINE (cos): Ratio of the length of the adjacent side to the length of the hypotenuse. | TANGENT (tan): Ratio of the length of the opposite side to the length of the adjacent side.

What's Next

What to Learn Next

Now that you can find unknown sides, you should learn how to find unknown angles using inverse trigonometric ratios. This will complete your understanding of solving right-angled triangles and open up even more exciting applications!