Overview of Trigonometric Inverse of Triangle in JEE Maths

Trigonometric inverse functions are essential tools for solving triangles in JEE Maths. They help calculate angles of a triangle when sides are known, or vice versa, using sin⁻¹, cos⁻¹, and tan⁻¹ functions. These concepts are crucial for coordinate geometry, calculus, and complex problem-solving. At StudentBro.in, we provide a comprehensive guide covering definitions, formulas, examples, and practice problems to help students master inverse trigonometry in triangles for JEE Main & Advanced exams.

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Definition of Trigonometric Inverse of Triangle

The inverse trigonometric function is used to find the angle of a triangle when the value of sine, cosine, or tangent is known. Notations: sin⁻¹ x (arcsin x), cos⁻¹ x (arccos x), tan⁻¹ x (arctan x). For example, if sin θ = 1/2 → θ = sin⁻¹(1/2) = 30°. These functions are fundamental for solving triangles using trigonometric ratios.

Law of Sines

For a triangle ABC with sides a, b, c opposite to angles A, B, C:

a / sin A = b / sin B = c / sin C

Using the inverse sine function, angles can be calculated when sides are known: A = sin⁻¹((a * sin B) / b). Essential for solving oblique triangles in JEE Maths.

Law of Cosines

For triangle ABC:

c² = a² + b² − 2ab cos C

Angle C can be found using inverse cosine: C = cos⁻¹((a² + b² − c²)/(2ab)). Useful when two sides and included angle or three sides are given.

Law of Tangents

For triangle ABC:

(a − b)/(a + b) = tan((A − B)/2) / tan((A + B)/2)

Using inverse tangent, angles of triangle can be calculated.

Inverse Trigonometric Functions

• sin⁻¹ x (arcsin x): Gives angle θ such that sin θ = x, θ ∈ [−90°, 90°]
• cos⁻¹ x (arccos x): Gives angle θ such that cos θ = x, θ ∈ [0°, 180°]
• tan⁻¹ x (arctan x): Gives angle θ such that tan θ = x, θ ∈ (−90°, 90°)

These inverse functions are used to find unknown angles in triangles when sides are known.

Solving Right-Angled Triangles

Use basic trigonometric ratios and their inverse functions:

θ = sin⁻¹(opposite / hypotenuse)

θ = cos⁻¹(adjacent / hypotenuse)

θ = tan⁻¹(opposite / adjacent)

Step-by-step method:

1. Identify sides of the triangle
2. Apply the appropriate ratio
3. Use inverse trigonometric function to calculate angle

Solving Oblique Triangles

Step 1: Use Law of Sines or Law of Cosines to find unknown sides or angles.
Step 2: Apply inverse trigonometric function to find angles.
Step 3: Verify triangle property: A + B + C = 180°.

Example: Triangle with sides a = 7, b = 8, c = 9. Find angle A: A = cos⁻¹((b² + c² − a²)/(2bc)).

Properties and Tips for Inverse Trigonometry in Triangles

• Check triangle inequality before solving
• Always verify angles sum to 180°
• Use Law of Sines when two sides and one angle or two angles and one side are known
• Use Law of Cosines when three sides or two sides and included angle are known
• Use calculator in degree mode for JEE Main & Advanced accuracy

Applications in JEE Maths

• Solve triangle problems in coordinate geometry
• Find distances and heights using triangles
• Solve physics problems involving triangles
• Apply in complex numbers, 3D geometry, and vectors
• Essential for speed and accuracy in JEE exams

StudentBro.in JEE Maths Trigonometric Inverse Section

At StudentBro.in, students can access:

• Step-by-step explanations for sin⁻¹, cos⁻¹, tan⁻¹ in triangles
• Worked examples for right-angled and oblique triangles
• Practice questions with solutions and shortcuts
• Tips for quick angle calculation using inverse trigonometry
• Revision notes for last-minute preparation

Practice Questions and Sample Problems

1. Solve a triangle with sides a = 6, b = 8, c = 10 using Law of Cosines
2. Find angle A if sin A = 3/5
3. Triangle with sides 7, 24, 25. Find the largest angle
4. Solve an oblique triangle with sides a = 9, b = 12 and angle C = 60°
5. Verify Law of Sines for triangle with sides 5, 6, 7

Conclusion

Trigonometric Inverse of Triangle is a vital topic in JEE Maths, enabling students to solve triangles, calculate unknown angles, and apply laws systematically. Understanding inverse functions, Law of Sines, Law of Cosines, and Law of Tangents allows students to tackle complex triangle problems efficiently. At StudentBro.in, we provide a complete guide from basics to advanced problem-solving, making inverse trigonometry in triangles simple, practical, and exam-oriented.

Mastering this topic ensures students can solve all types of triangle problems confidently in both JEE Main and Advanced exams.