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Trigonometry 1b - Solved problems related to Basic Trigonometric Ratios, Degrees and Radians, Domains and ranges, Basic Identities, Compound Angle Formulas, Double and Triple Angle Formulas

Inverse Trigonometric Ratios

Target Audience: High School Students, College Freshmen and Sophomores, Class 11/12 Students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE, Anyone else who needs this Tutorial as a reference!

Solved Problems

In this tutorial, we'll try to apply the basics you learnt about Trigonometric ratios, identities and compound angle formulas.
1. Convert 6 radians into degree measures.
2. The angles of a triangle are in A. P. and the number of grades in the least is to the number  of radians in the greatest as 40 : 7 π, find the angles in degrees. 
3. Assuming the average distance of the earth from the sun to be 92500000 miles, and the angle subtended by the sun at the eye of a person on the earth to be 32', find the sun's diameter.  
4. If cot x = – 5/12 , x lies in second quadrant, find the values of other five trigonometric functions.
5. Prove that (cosec a - sin a) (sec a - cos a) (tan a + cota) = 1. 
6. Find the values of sin 75° and cos 75°.
7. Simplify the expression ((cosx-cos3x)(sin8x+sin2x)) / ((sin5x-sinx)(cos4x-cos6x))
8. Given cos 330o= 30.5/2 ,find the values of sin 165° and cos165°. 
9. Given tan15o=2-30.5, find tan (7.5 degrees).
10. When A + B + C = 180°, show that sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C.
11. Find the maximum and minimum values of 4 cos x + 3 sin x on the interval 0 ≤ x 360o.
12. Construct a graph of each of the following . (a) y = 3 sin x + 1 ( c ) y = cos x + 2 (b) y = sin x - 2 ( d ) y = 12 cos x – 1
13. Solve the equation 3cosx + 4sinx=5, for principal solutions
14. What is the most general value of  θ which satisfies both of the equations sin θ = - 12  and tan θ =  13 ?
15. Solve the equation 2sin 2 x+3cosx+l=0. 
16. Solve the system
r sinθ = 3, r = 4(1 + sinθ)
for r > 0 and 0 θ < 2π
17. Show that the values of the trigonometric functions of an angle θ do not depend on the choice of the point P selected on the terminal side of the angle.
18. Let a and b be nonnegative real numbers.
(a) Prove that there is a real number x such that sin x +a cos x = b if and only if a2 b2 + 1 ≥ 0.
(b) If sin x + a cos x = b, express |a sin x − cos x| in terms of a and b.
19. Let f be an odd function defined on the real numbers such that for x ≥ 0, f (x) = 3 sin x + 4 cos x. Find f (x) for x < 0.
20. Let x, y, z be positive real numbers such that x+y+z = 1. Determine the minimum value of 1/x+4/y+9/z.
21. Let a and b be real numbers such that
sin a + sin b =22,
cos a + cos b =62.
Evaluate sin(a + b).
22. Region R contains all the points (x, y) such that x2 + y2 ≤ 100 and sin(x + y) ≥ 0. Find the area of region R.
23. Express sin(x y) + sin(y z) + sin(z x) as a monomial.
24. Prove that (1+asinx) (1+bcosx) ≥ (1+2ab)2 for all real numbers a, b, x with a, b ≥ 0 and 0 < x < π2
25. Prove that cos 1o is an irrational number.
26. Let a and b be real numbers in the interval [0, π/2]. Prove that sin6 a + 3 sin2 a cos2 b + cos6 b = 1 if and only if a = b.

    • Complete tutorial with examples, problems and solutions. You can view or download the PDF form of the tutorial here :

MCQ Quiz #1: Basics of Trigonometric Ratios

MCQ Quiz: Basics of Trigonometric Ratios

In case you'd like to take a look at other Trigonometry tutorials : 

Trigonometry 1a ( Introduction to Trigonometry - Definitions, Formulas ) Introducing trigonometric ratios, plots of trigonometric functions, compound angle formulas. Domains and ranges of trigonometric functions, monotonicity of trigonometric functions quadrant wise. Formulas for double and triple angle ratios.

Trigonometry 1b ( Tutorial with solved problems based on Trigonometric ratios ) Problems based on the concepts introduced above.
Trigonometry 2a ( Basic concepts related to Heights and Distances ) Applying trigonometry to problems involving heights and distances. Angles of elevation and depression. Sine and Cosine rule, half angle formulas. Circumradius, inradius and escribed radius. Circumcentre, incentre, centroid and median of a triangle.

Trigonometry 2b ( Tutorial with solved problems related to Heights and Distances and other applications of Trigonometry ) - Problems based on the concepts introduced above.

Trigonometry 3a ( Introducing Inverse Trigonometric Ratios)
Inverse trigonometric ratios - their domains, ranges and plots. 

Trigonometry 3b ( Tutorial with solved problems related to inverse trigonometric ratios )- Problems related to inverse trigonometric ratios. 

Trigonometry 4 ( A tutorial on solving trigonometric equations )- Solving trigonometric equations. Methods and transformations frequently used in solving such equations.