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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).
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.
Prove that sin²θ + cos²θ = 1 for any acute angle θ.
Use the Pythagorean theorem in a right-angled triangle to relate the sides.
Explain the significance of trigonometric ratios in real-life scenarios with examples.
Think about how angles and distances are measured in various fields.
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.
Calculate the trigonometric ratios for a 45° angle using a right-angled triangle.
Consider an isosceles right-angled triangle with equal legs.
Discuss the importance of the Pythagorean theorem in trigonometry.
Relate the Pythagorean theorem to the derivation of trigonometric identities.
Explain how to find the value of trigonometric ratios for 30° and 60° angles.
Use an equilateral triangle divided into two right-angled triangles.
Describe the concept of complementary angles in trigonometry with examples.
Consider how the sides of a right-angled triangle relate to complementary angles.
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.
Explain the use of trigonometric identities in simplifying expressions.
Think about how identities can replace one trigonometric function with another.
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.
Questions
Prove that sin²θ + cos²θ = 1 using a right-angled triangle.
Start with Pythagoras theorem and divide by the square of the hypotenuse.
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.
Compare and contrast sine and cosine functions with respect to their graphs and properties.
Focus on their maximum, minimum values, and phase difference.
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.
Explain why the value of tan90° is undefined.
Consider the definition of tanθ and the value of cosθ at 90°.
Derive the expression for cotθ in terms of sinθ and cosθ.
Start with the definition of tanθ and take its reciprocal.
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.
Prove that (1 + tan²θ) = sec²θ using the fundamental identity.
Divide the fundamental trigonometric identity by cos²θ.
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.
Compare the trigonometric ratios of complementary angles.
Consider the definitions of the trigonometric ratios in a right-angled triangle with angles A and B.
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.
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.
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.
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.
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.
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.
If sin θ + cos θ = √2 cos θ, prove that cot θ = √2 + 1.
Remember that cot θ = cos θ / sin 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.
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.
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.
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.
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.
