Trigonometry Solved Examples (Class 10 Maths)
Trigonometry deals with the relationships between sides and angles in triangles. The trigonometric ratios sine, cosine, and tangent are fundamental tools f
TL;DR: Trigonometry deals with the relationships between sides and angles in triangles. The trigonometric ratios sine, cosine, and tangent are fundamental to…
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Trigonometry deals with the relationships between sides and angles in triangles. The trigonometric ratios sine, cosine, and tangent are fundamental tools f
Trigonometry — Solved Numerical Examples (Step by Step)
Example 1: In a right triangle, the opposite side to an angle is 3 cm and the hypotenuse is 5 cm. Find the sine of that angle.
Solution: Given:
Opposite side = 3 cm
Hypotenuse = 5 cm
Using the definition: sin(angle) = opposite / hypotenuse
sin(angle) = 3/5 = 0.6
Example 2: Find the value of cos(60 degrees).
Solution: The angle 60 degrees is a standard angle.
From the standard trigonometric values:
cos(60 degrees) = 1/2 = 0.5
Example 3: A ladder 10 m long is leaned against a wall such that it makes an angle of 60 degrees with the ground. Find the height at which the ladder touches the wall.
Solution: Given:
Length of ladder (hypotenuse) = 10 m
Angle with ground = 60 degrees
We need to find the height (opposite to the angle)
Using: sin(60) = height / hypotenuse
sin(60) = height / 10
sqrt(3)/2 = height / 10
height = 10 × sqrt(3)/2 = 5sqrt(3) = 8.66 m (approximately)
Example 4: The angle of elevation to the top of a building from a point 50 m away is 30 degrees. Find the height of the building.
Solution: Given:
Horizontal distance = 50 m
Angle of elevation = 30 degrees
Using: tan(angle) = height / distance
tan(30) = height / 50
1/sqrt(3) = height / 50
height = 50 / sqrt(3) = 50sqrt(3) / 3 = 28.87 m (approximately)
Example 5: In a right triangle, the adjacent side is 4 cm and the hypotenuse is 5 cm. Find the cosine of that angle.
Solution: Given:
Adjacent side = 4 cm
Hypotenuse = 5 cm
Using the definition: cos(angle) = adjacent / hypotenuse
cos(angle) = 4/5 = 0.8
Example 6: Find the value of tan(45 degrees).
Solution: The angle 45 degrees is a standard angle in a 45-45-90 triangle.
In such a triangle, the two legs are equal.
Using: tan(45) = opposite / adjacent = 1/1 = 1
Example 7: Find the angle whose sine is 1/2.
Solution: We need to find angle A such that sin(A) = 1/2
From standard trigonometric values:
sin(30 degrees) = 1/2
Therefore, A = 30 degrees (in the range 0 to 90 degrees)
Example 8: In a right triangle, if tan(A) = 3/4, find sin(A) and cos(A).
Solution: Given: tan(A) = 3/4 = opposite/adjacent
So opposite = 3, adjacent = 4
Using Pythagoras theorem:
hypotenuse^2 = 3^2 + 4^2 = 9 + 16 = 25
hypotenuse = 5
sin(A) = opposite / hypotenuse = 3/5
cos(A) = adjacent / hypotenuse = 4/5
Tips
- Remember SOH-CAH-TOA: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. This helps recall which ratio to use.
- Learn the standard angles (0, 30, 45, 60, 90 degrees) and their sine, cosine, and tangent values as they appear frequently in problems.
- In angle of elevation and depression problems, carefully identify which side is opposite, adjacent, or hypotenuse relative to the given angle.
Frequently Asked Questions
What is the difference between angle of elevation and angle of depression?
Angle of elevation is measured upward from the horizontal line to an object above the observer. Angle of depression is measured downward from the horizontal line to an object below the observer. Both are measured from the horizontal.
Why are sine and cosine values always between 0 and 1?
Because sine and cosine are ratios of a side to the hypotenuse in a right triangle. Since the hypotenuse is always the longest side, the ratio of any side to the hypotenuse is always less than 1.
More Maths Solved Examples
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