Introduction to Trigonometry — Important Questions
25 questions
With answersCBSE format
SUMMARY: This chapter introduces the basic concepts of trigonometry, focusing on the relationships between the angles and sides of right-angled triangles. KEY TOPICS: trigonometric ratios, sine, cosine, tangent, complementary angles, trigonometric identities, right-angled triangle, angle of elevation, angle of depression, applications of trigonometry.
In a right-angled triangle, if the length of the opposite side is 3 cm and the length of the hypotenuse is 5 cm, what is the value of sin(θ)?
A0.6
B0.5
C0.8
D0.4
Check answerHide answer
Correct answer: Option 1 — 0.6
Q21 Mark
Which of the following is the correct trigonometric identity?
Asin²(θ) + cos²(θ) = 1
Btan(θ) = sin(θ) + cos(θ)
Ccot(θ) = sin(θ) × cos(θ)
Dsec(θ) = 1 - cos(θ)
Check answerHide answer
Correct answer: Option 1 — sin²(θ) + cos²(θ) = 1
Q31 Mark
If tan(θ) = 3/4, what is the value of cos(θ)?
A4/5
B3/5
C5/4
D3/4
Check answerHide answer
Correct answer: Option 1 — 4/5
Q41 Mark
In a right triangle, if the angle of elevation from point A to the top of a tree is 30°, and the distance from point A to the base of the tree is 10 m, what is the height of the tree?
A5√3 m
B10 m
C5 m
D10√3 m
Check answerHide answer
Correct answer: Option 3 — 5 m
Q51 Mark
If the angle of depression from the top of a tower is 45° to a point on the ground, what is the relationship between the height of the tower and the distance from the base of the tower to the point on the ground?
AHeight = Distance
BHeight > Distance
CHeight < Distance
DHeight = 2 × Distance
Check answerHide answer
Correct answer: Option 1 — Height = Distance
Short Answer Questions5 questions
Q63 Marks
Define the sine, cosine, and tangent ratios in a right-angled triangle. How are they related to the sides of the triangle?
View sample solutionHide solution
In a right-angled triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse. The cosine is the ratio of the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side.
Q73 Marks
What are complementary angles? Provide an example using trigonometric ratios.
View sample solutionHide solution
Complementary angles are two angles whose sum is 90 degrees. For example, if angle A is 30 degrees, then angle B is 60 degrees. The sine of angle A (sin 30°) equals the cosine of angle B (cos 60°).
Q83 Marks
Explain the concept of angle of elevation and provide a real-life example where it can be applied.
View sample solutionHide solution
The angle of elevation is the angle formed between the horizontal line and the line of sight to an object above the horizontal. For example, when standing on the ground and looking up at the top of a building, the angle formed is the angle of elevation. This concept is often used in surveying and architecture.
Q93 Marks
Using trigonometric identities, prove that sin²θ + cos²θ = 1.
View sample solutionHide solution
To prove sin²θ + cos²θ = 1, we start with a right triangle where the hypotenuse is 1. By definition, sinθ = opposite/hypotenuse and cosθ = adjacent/hypotenuse. Thus, sin²θ + cos²θ = (opposite² + adjacent²)/hypotenuse² = 1² = 1, proving the identity.
Q103 Marks
A ladder is leaning against a wall, forming an angle of 75° with the ground. If the foot of the ladder is 2 meters away from the wall, find the height at which the ladder touches the wall using trigonometric ratios.
View sample solutionHide solution
Using the tangent ratio, tan(75°) = height / 2. Therefore, height = 2 * tan(75°). Calculating this gives the height at which the ladder touches the wall as approximately 7.73 meters.
Long Answer Questions5 questions
Q116 Marks
Define the trigonometric ratios for a right-angled triangle. Given a right triangle with an angle θ, express the sine, cosine, and tangent ratios in terms of the lengths of the sides of the triangle. Provide an example with specific side lengths.
View sample solutionHide solution
In a right-angled triangle, the trigonometric ratios are defined as follows: the sine of angle θ (sin θ) is the ratio of the length of the opposite side to the length of the hypotenuse; the cosine of angle θ (cos θ) is the ratio of the length of the adjacent side to the length of the hypotenuse; and the tangent of angle θ (tan θ) is the ratio of the length of the opposite side to the length of the adjacent side. For example, consider a right triangle where the length of the opposite side is 3 units, the adjacent side is 4 units, and the hypotenuse is 5 units. Then, sin θ = 3/5, cos θ = 4/5, and tan θ = 3/4.
Q126 Marks
Explain the concept of complementary angles in trigonometry. How do the trigonometric ratios of an angle relate to those of its complementary angle? Provide a mathematical proof for one of the trigonometric identities involving complementary angles.
View sample solutionHide solution
Complementary angles are two angles whose measures add up to 90 degrees. In trigonometry, the sine of an angle is equal to the cosine of its complementary angle, and vice versa. This relationship can be expressed as sin(θ) = cos(90° - θ) and cos(θ) = sin(90° - θ). To prove this, consider a right triangle where one angle is θ and the other is 90° - θ. By definition, sin(θ) = opposite/hypotenuse and cos(90° - θ) = adjacent/hypotenuse. Since the opposite side of angle θ becomes the adjacent side of angle (90° - θ), we have sin(θ) = cos(90° - θ), thus proving the identity.
Q136 Marks
A person is standing 50 meters away from the base of a tower. If the angle of elevation from the ground to the top of the tower is 30 degrees, calculate the height of the tower. Show your calculations and explain the steps involved.
View sample solutionHide solution
To find the height of the tower, we can use the tangent function, which relates the angle of elevation to the opposite side (height of the tower) and the adjacent side (distance from the tower). Let h be the height of the tower. We have tan(30°) = h/50. The value of tan(30°) is √3/3. Therefore, we can set up the equation: √3/3 = h/50. Solving for h gives us h = 50 * (√3/3) = 50√3/3 meters. Thus, the height of the tower is approximately 28.87 meters.
Q146 Marks
Using trigonometric identities, prove that 1 + tan²(θ) = sec²(θ). Provide a detailed explanation of each step in your proof.
View sample solutionHide solution
To prove the identity 1 + tan²(θ) = sec²(θ), we start with the definition of the tangent and secant functions. We know that tan(θ) = sin(θ)/cos(θ) and sec(θ) = 1/cos(θ). Therefore, tan²(θ) = sin²(θ)/cos²(θ). Substituting this into the left side of the equation gives us 1 + sin²(θ)/cos²(θ). To combine these terms, we can express 1 as cos²(θ)/cos²(θ), leading to (cos²(θ) + sin²(θ))/cos²(θ). By the Pythagorean identity, we know that sin²(θ) + cos²(θ) = 1, so we can simplify this to 1/cos²(θ), which is equal to sec²(θ). Thus, we have proven that 1 + tan²(θ) = sec²(θ).
Q156 Marks
Describe the applications of trigonometry in real-life scenarios. Provide at least two examples where trigonometric concepts are used to solve practical problems.
View sample solutionHide solution
Trigonometry has numerous applications in real life, especially in fields such as architecture, engineering, and navigation. One example is in construction, where trigonometric ratios are used to determine the height of structures. For instance, if a surveyor measures the angle of elevation to the top of a building from a certain distance, they can use trigonometric functions to calculate the building's height. Another example is in navigation, where trigonometry helps in determining the position of a ship or an aircraft. By measuring angles and distances from known points, navigators can accurately plot their course using sine, cosine, and tangent ratios to calculate distances and angles.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): The sine of an angle in a right-angled triangle is equal to the ratio of the length of the opposite side to the hypotenuse.
Reason (R): The cosine of an angle is defined as the ratio of the adjacent side to the hypotenuse.
Show explanationHide explanation
Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q171 Mark
Assertion (A): The tangent of an angle is the ratio of the sine of the angle to the cosine of the angle.
Reason (R): The tangent function is defined only for acute angles in a right triangle.
Show explanationHide explanation
Correct answer: Option 3 —
A is true, but R is false.
Q181 Mark
Assertion (A): If two angles are complementary, then the sine of one angle is equal to the cosine of the other.
Reason (R): Complementary angles always add up to 90 degrees, which is the basis for the sine and cosine relationship.
Show explanationHide explanation
Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q191 Mark
Assertion (A): In a right-angled triangle, the angle of elevation is measured from the horizontal line to the line of sight.
Reason (R): The angle of depression is measured from the line of sight to the horizontal line.
Show explanationHide explanation
Correct answer: Option 2 —
Both A and R are true, but R is not the correct explanation of A.
Q201 Mark
Assertion (A): The trigonometric identity sin²θ + cos²θ = 1 holds true for all angles.
Reason (R): This identity is only applicable to angles in right-angled triangles.
Show explanationHide explanation
Correct answer: Option 3 —
A is true, but R is false.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: The sine of an angle in a right-angled triangle is the ratio of the length of the opposite side to the hypotenuse.
Statement 2: The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.
Show answerHide answer
Correct answer: Option 1 —
Both statements are true.
Q221 Mark
Statement 1: In a right-angled triangle, the tangent of an angle is equal to the sine of that angle divided by the cosine of that angle.
Statement 2: The angle of elevation is always greater than the angle of depression.
Show answerHide answer
Correct answer: Option 3 —
Only Statement 2 is true.
Q231 Mark
Statement 1: If two angles are complementary, then the sine of one angle is equal to the cosine of the other angle.
Statement 2: The trigonometric identity sin²θ + cos²θ = 1 holds true for all angles θ.
Show answerHide answer
Correct answer: Option 1 —
Both statements are true.
Q241 Mark
Statement 1: The angle of depression is measured from the horizontal line down to the object.
Statement 2: The tangent of an angle can be found using the formula tan(θ) = opposite/adjacent.
Show answerHide answer
Correct answer: Option 1 —
Both statements are true.
Q251 Mark
Statement 1: In a right-angled triangle, the hypotenuse is always the longest side.
Statement 2: The sine ratio can be applied to any triangle, not just right-angled triangles.
Show answerHide answer
Correct answer: Option 2 —
Only Statement 1 is true.
