Worksheet
Explore the basics of trigonometry, including angles, triangles, and the fundamental trigonometric ratios: sine, cosine, and tangent.
Introduction to Trigonometry - Practice Worksheet
Strengthen your foundation with key concepts and basic applications.
This worksheet covers essential long-answer questions to help you build confidence in Introduction to Trigonometry from Mathematics for Class X (Mathematics).
Basic comprehension exercises
Strengthen your understanding with fundamental questions about the chapter.
Questions
Define trigonometric ratios and explain how they are derived in a right-angled triangle.
Recall the definitions of sine, cosine, and tangent in terms of the sides of a right-angled triangle.
Solution
Trigonometric ratios are ratios of the sides of a right-angled triangle relative to one of its acute angles. They are derived by comparing the lengths of two sides of the triangle. The six trigonometric ratios are sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot). For an angle θ in a right-angled triangle ABC, right-angled at B, sin θ = opposite side/hypotenuse = BC/AC, cos θ = adjacent side/hypotenuse = AB/AC, and tan θ = opposite side/adjacent side = BC/AB. The reciprocals of these ratios give cosec θ = AC/BC, sec θ = AC/AB, and cot θ = AB/BC. These ratios are fundamental in solving problems involving right-angled triangles and have applications in various fields such as physics, engineering, and astronomy.
Prove that sin²θ + cos²θ = 1 for any acute angle θ.
Use the Pythagorean theorem in a right-angled triangle to relate the sides.
Solution
To prove the identity sin²θ + cos²θ = 1, consider a right-angled triangle ABC with angle θ at A. By definition, sin θ = BC/AC and cos θ = AB/AC. Squaring both and adding gives sin²θ + cos²θ = (BC/AC)² + (AB/AC)² = (BC² + AB²)/AC². According to the Pythagorean theorem, BC² + AB² = AC². Therefore, sin²θ + cos²θ = AC²/AC² = 1. This identity is fundamental in trigonometry and is used to derive other identities and solve trigonometric equations.
Explain the significance of trigonometric ratios in real-life scenarios with examples.
Think about how angles and distances are measured in various fields.
Solution
Trigonometric ratios have wide applications in real-life scenarios. For example, in architecture, they are used to calculate the height of buildings or the angle of a roof. In navigation, they help in determining the direction and distance between two points. Astronomers use these ratios to calculate the distances between celestial bodies. Another example is in the field of engineering, where trigonometric ratios are essential in designing bridges, ramps, and other structures to ensure stability and safety. These ratios are also used in computer graphics to create realistic animations and in physics to analyze wave patterns.
Derive the relationship between tan θ and the other trigonometric ratios.
Express tan θ in terms of sin θ and cos θ by dividing the two ratios.
Solution
The tangent of an angle θ, tan θ, is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. It can also be expressed in terms of sine and cosine ratios. Since sin θ = opposite/hypotenuse and cos θ = adjacent/hypotenuse, dividing sin θ by cos θ gives (opposite/hypotenuse)/(adjacent/hypotenuse) = opposite/adjacent = tan θ. Therefore, tan θ = sin θ/cos θ. This relationship is useful in simplifying trigonometric expressions and solving equations.
Calculate the trigonometric ratios for a 45° angle using a right-angled triangle.
Consider an isosceles right-angled triangle with equal legs.
Solution
For a 45° angle in a right-angled triangle, the two non-right angles are equal, making the triangle isosceles with the two legs of equal length. Let's assume each leg is of length 1. The hypotenuse can be found using the Pythagorean theorem: √(1² + 1²) = √2. Therefore, sin 45° = opposite/hypotenuse = 1/√2, cos 45° = adjacent/hypotenuse = 1/√2, and tan 45° = opposite/adjacent = 1/1 = 1. The reciprocal ratios are cosec 45° = √2, sec 45° = √2, and cot 45° = 1. These values are standard and frequently used in trigonometric calculations.
Discuss the importance of the Pythagorean theorem in trigonometry.
Relate the Pythagorean theorem to the derivation of trigonometric identities.
Solution
The Pythagorean theorem is fundamental in trigonometry as it establishes a relationship between the sides of a right-angled triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is used to derive the basic trigonometric identity sin²θ + cos²θ = 1, which is the foundation for many other identities. It also allows for the calculation of one side of a triangle if the other two are known, facilitating the determination of trigonometric ratios. Without the Pythagorean theorem, many trigonometric proofs and applications would not be possible.
Explain how to find the value of trigonometric ratios for 30° and 60° angles.
Use an equilateral triangle divided into two right-angled triangles.
Solution
To find the trigonometric ratios for 30° and 60° angles, consider an equilateral triangle with each side of length 2. When divided into two right-angled triangles, each has a hypotenuse of 2, a base of 1, and a height of √3. For the 30° angle, opposite side = 1, adjacent side = √3, and hypotenuse = 2. Thus, sin 30° = 1/2, cos 30° = √3/2, and tan 30° = 1/√3. For the 60° angle, opposite side = √3, adjacent side = 1, and hypotenuse = 2. Thus, sin 60° = √3/2, cos 60° = 1/2, and tan 60° = √3. These values are essential for solving problems involving these angles.
Describe the concept of complementary angles in trigonometry with examples.
Consider how the sides of a right-angled triangle relate to complementary angles.
Solution
In trigonometry, two angles are complementary if their sum is 90°. The trigonometric ratios of complementary angles are related. For example, sin(90° - θ) = cos θ, cos(90° - θ) = sin θ, and tan(90° - θ) = cot θ. This is because the opposite side for angle θ becomes the adjacent side for its complement (90° - θ), and vice versa. For instance, in a right-angled triangle with angles 30° and 60°, sin 30° = cos 60° = 1/2, and cos 30° = sin 60° = √3/2. These relationships simplify calculations and are used in solving trigonometric problems.
Solve for θ if sin θ = cos θ, where θ is an acute angle.
Divide both sides by cos θ to express the equation in terms of tan θ.
Solution
To solve sin θ = cos θ for an acute angle θ, we can use the identity tan θ = sin θ/cos θ. Dividing both sides of the equation by cos θ gives tan θ = 1. The acute angle whose tangent is 1 is 45°. Therefore, θ = 45°. This can also be verified by considering a right-angled triangle with θ = 45°, where the opposite and adjacent sides are equal, making sin θ = cos θ = 1/√2. This solution is unique for acute angles and is a common trigonometric equation.
Explain the use of trigonometric identities in simplifying expressions.
Think about how identities can replace one trigonometric function with another.
Solution
Trigonometric identities are equations that hold true for all values of the variables involved. They are used to simplify complex trigonometric expressions, making them easier to solve or integrate. For example, the identity sin²θ + cos²θ = 1 can be used to express sin²θ as 1 - cos²θ or vice versa. Similarly, 1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ are used to convert between different trigonometric functions. These identities are essential in calculus, physics, and engineering, where they help in simplifying integrals, solving differential equations, and analyzing wave functions.
Introduction to Trigonometry - Mastery Worksheet
Advance your understanding through integrative and tricky questions.
This worksheet challenges you with deeper, multi-concept long-answer questions from Introduction to Trigonometry to prepare for higher-weightage questions in Class X Mathematics.
Intermediate analysis exercises
Deepen your understanding with analytical questions about themes and characters.
Questions
Prove that sin²θ + cos²θ = 1 using a right-angled triangle.
Start with Pythagoras theorem and divide by the square of the hypotenuse.
Solution
Consider a right-angled triangle ABC with angle θ at A. Let AB = adjacent side, BC = opposite side, and AC = hypotenuse. By Pythagoras theorem, AB² + BC² = AC². Dividing both sides by AC², we get (AB/AC)² + (BC/AC)² = 1. But AB/AC = cosθ and BC/AC = sinθ. Therefore, cos²θ + sin²θ = 1.
If tanθ = 3/4, find the values of sinθ and cosθ without using Pythagoras theorem.
Use the definition of tanθ to express sides in terms of a variable k.
Solution
Given tanθ = 3/4 = opposite/adjacent. Let opposite = 3k and adjacent = 4k. Then, hypotenuse = √(3k)² + (4k)² = 5k. Therefore, sinθ = opposite/hypotenuse = 3/5 and cosθ = adjacent/hypotenuse = 4/5.
Compare and contrast sine and cosine functions with respect to their graphs and properties.
Focus on their maximum, minimum values, and phase difference.
Solution
Both sine and cosine functions are periodic with a period of 2π. The sine function starts at 0, reaches 1 at π/2, back to 0 at π, -1 at 3π/2, and 0 at 2π. The cosine function starts at 1, decreases to 0 at π/2, -1 at π, back to 0 at 3π/2, and 1 at 2π. Both have an amplitude of 1 and are phase-shifted versions of each other by π/2.
A ladder leans against a wall making an angle of 60° with the ground. If the foot of the ladder is 2m away from the wall, find the length of the ladder.
Use the cosine of the angle to relate the known side to the hypotenuse.
Solution
The scenario forms a right-angled triangle with the ladder as the hypotenuse. The angle between the ground and the ladder is 60°, and the adjacent side (distance from the wall) is 2m. Using cosθ = adjacent/hypotenuse, cos60° = 2/length of ladder. Therefore, length of ladder = 2/cos60° = 2/(1/2) = 4m.
Explain why the value of tan90° is undefined.
Consider the definition of tanθ and the value of cosθ at 90°.
Solution
Tanθ is defined as sinθ/cosθ. At θ=90°, cosθ=0, making the denominator zero. Division by zero is undefined in mathematics, hence tan90° is undefined.
Derive the expression for cotθ in terms of sinθ and cosθ.
Start with the definition of tanθ and take its reciprocal.
Solution
Cotθ is the reciprocal of tanθ. Since tanθ = sinθ/cosθ, cotθ = cosθ/sinθ.
If secθ = 13/5, find the value of tanθ without finding θ.
Use the definition of secθ to express sides in terms of a variable k and then find the opposite side using Pythagoras theorem.
Solution
Given secθ = 13/5 = hypotenuse/adjacent. Let hypotenuse = 13k and adjacent = 5k. Then, opposite = √(13k)² - (5k)² = 12k. Therefore, tanθ = opposite/adjacent = 12/5.
Prove that (1 + tan²θ) = sec²θ using the fundamental identity.
Divide the fundamental trigonometric identity by cos²θ.
Solution
Starting with the fundamental identity sin²θ + cos²θ = 1, divide both sides by cos²θ to get (sin²θ/cos²θ) + (cos²θ/cos²θ) = 1/cos²θ. This simplifies to tan²θ + 1 = sec²θ, since tanθ = sinθ/cosθ and secθ = 1/cosθ.
A pole casts a shadow of length 20m when the sun's elevation is 30°. Find the height of the pole.
Use the tangent of the angle of elevation to relate the height of the pole to the length of the shadow.
Solution
The scenario forms a right-angled triangle with the pole height as the opposite side to the angle of elevation and the shadow as the adjacent side. Using tanθ = opposite/adjacent, tan30° = height/20. Therefore, height = 20 * tan30° = 20 * (1/√3) ≈ 11.55m.
Compare the trigonometric ratios of complementary angles.
Consider the definitions of the trigonometric ratios in a right-angled triangle with angles A and B.
Solution
For complementary angles A and B (A + B = 90°), sinA = cosB, cosA = sinB, tanA = cotB, cotA = tanB, secA = cosecB, and cosecA = secB. This is because the side opposite to angle A is adjacent to angle B and vice versa in a right-angled triangle.
Introduction to Trigonometry - Challenge Worksheet
Push your limits with complex, exam-level long-form questions.
The final worksheet presents challenging long-answer questions that test your depth of understanding and exam-readiness for Introduction to Trigonometry in Class X.
Advanced critical thinking
Test your mastery with complex questions that require critical analysis and reflection.
Questions
Prove that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides using trigonometric identities.
Consider how the definitions of sine and cosine relate to the sides of a right-angled triangle.
Solution
Start with the trigonometric identity sin²θ + cos²θ = 1. Multiply both sides by the square of the hypotenuse to relate it to the other two sides.
If tan θ = 1/√3, find the value of (sin θ + cos θ) / (sin θ - cos θ) without using a calculator.
Recall that tan θ = opposite/adjacent and use the Pythagorean theorem to find the hypotenuse.
Solution
First, determine the values of sin θ and cos θ using the given tan θ. Then, substitute these values into the expression to find the result.
Evaluate the expression: sin 30° cos 60° + cos 30° sin 60°. What trigonometric identity does this resemble?
Think about the angle addition formulas in trigonometry.
Solution
Calculate each trigonometric ratio using standard values and then perform the addition. Compare the result to known trigonometric identities.
A ladder leans against a wall making an angle of 60° with the ground. If the foot of the ladder is 2m away from the wall, find the length of the ladder.
The cosine of an angle in a right-angled triangle is adjacent/hypotenuse.
Solution
Use the cosine of the angle to relate the distance from the wall to the length of the ladder.
Prove the identity: (1 + cot θ - cosec θ)(1 + tan θ + sec θ) = 2.
Express all terms in terms of sine and cosine to simplify the expression.
Solution
Expand the left-hand side using trigonometric identities and simplify to show it equals 2.
If sin θ + cos θ = √2 cos θ, prove that cot θ = √2 + 1.
Remember that cot θ = cos θ / sin θ.
Solution
Divide both sides of the equation by sin θ to relate it to cot θ and then solve for cot θ.
Find the value of θ if 2 sin² θ - 3 sin θ + 1 = 0, where 0° ≤ θ ≤ 90°.
Treat sin θ as a variable and solve the quadratic equation.
Solution
Solve the quadratic equation in terms of sin θ and then find the corresponding angles within the given range.
Prove that (sin A + cosec A)² + (cos A + sec A)² = 7 + tan² A + cot² A.
Recall that cosec A = 1/sin A and sec A = 1/cos A.
Solution
Expand both squares, use trigonometric identities, and simplify to reach the right-hand side.
A tower stands vertically on the ground. From a point on the ground which is 20m away from the foot of the tower, the angle of elevation of the top of the tower is 60°. Find the height of the tower.
The tangent of an angle in a right-angled triangle is opposite/adjacent.
Solution
Use the tangent of the angle of elevation to relate the distance from the tower to its height.
Prove that the trigonometric ratios of an angle do not depend on the size of the right-angled triangle but only on the angle.
Use the property of similar triangles that corresponding sides are proportional.
Solution
Consider two similar right-angled triangles and show that the ratios of corresponding sides are equal for the same angle.
Explore the methods to solve a pair of linear equations in two variables, including graphical, substitution, elimination, and cross-multiplication techniques.
Explore the world of quadratic equations, learning to solve them using various methods like factoring, completing the square, and the quadratic formula.
A chapter that explores sequences where each term after the first is obtained by adding a constant difference, focusing on their properties, nth term, and sum formulas.
Explore the properties, types, and theorems related to triangles, including congruence and similarity, to solve geometric problems effectively.
Coordinate Geometry explores the relationship between algebra and geometry through the use of coordinate systems to represent geometric shapes and solve problems.
