How to Solve Trigonometric Function Related Problems

Trigonometry is a field of mathematics that explores the connections between triangle side lengths and angles.

There are six fundamental trigonometric functions in trigonometry. The functions in question are trigonometric ratios. The six fundamental trigonometric functions are the sine function, cosine function, secant function, co-secant function, tangent function, and co-tangent function. Trigonometric functions and identities are based on the ratio of sides of a right-angled triangle. The sine, cosine, tangent, secant, cosecant, and cotangent values are calculated using the perpendicular side, hypotenuse, and base of a right triangle in trigonometric formulas.

Using the sides of a right-angled triangle, we may get the values of the trig functions using specific formulae. We utilise the shortened version of these functions to create these formulae. Sine is shortened as sin, cosine is abbreviated as cos, tangent is shorthand as tan, secant is abbreviated as sec, and cotangent is abbreviated as cot. The following are the fundamental formulae for finding trigonometric functions:

• sin θ = Perpendicular/Hypotenuse

• cos θ = Base/Hypotenuse

• tan θ = Perpendicular/Base

• sec θ = Hypotenuse/Base

• cosec θ = Hypotenuse/Perpendicular

• cot θ = Base/Perpendicular

Sine and cosecant are reciprocals of each other, as we can see from the formulae above. Cosine and secant, as well as tangent and cotangent, are reciprocal pairings.

Now we are going to have some questions with their solutions so that the procedure to carry out any solution to trigonometric problems could be proceeded.

Determine the Value of SIN75°

Ans – We have to find the value of Sin75°.

Now we will use the formula 

Sin(A + B) = SinA. CosB + CosA.SinB.

take  A = 30° and B = 45°

Sin 75° = Sin(30° + 45°)

= Sin30°.Cos45° + Cos30°.Sin45°

= (1/2) (1/√2) + (√3/2) (1/√2)

= 1/2√2 + √3/2√2

= (√3 + 1) / 2√2

Determine the Value of the Given Trigonometric Functions, For 12Tanθ = 5.

Ans – Given 12Tanθ = 5, 

we get Tanθ = 5/12

we know , Tanθ = Perpendicular/Base = 5/12

from Pythagorean theorem we get:

Hypotenuse2 = Perpendicular2 + Base2

Hyp2 = 122 + 52

= 144 + 25

= 169

Hyp = 13

the required other trigonometric functions are :

Sinθ = Perp/Hyp = 5/13

Cosθ = Base/Hyp = 12/13

Cotθ = Base/Perp = 12/5

Secθ = Hyp/Base = 13/12

Cosecθ = Hyp/Perp = 13/5

Find the Product of all Trigonometric Functions

Ans – Since we know that,

reciprocal of sin x is cosec x 

reciprocal of cos x is sec x. 

Also, tan x can be expressed as the ratio of sin x and cos x

cot x can be expressed as the ratio of cos x and sin x. 

So, we get

sinx × cosx × tanx × cotx × secx × cosecx = sinx × cosx × (sinx/cosx) × (cosx/sinx) × (1/cosx) × (1/sinx)

= (sinx × cosx) / (sinx × cosx) × (sinx/cosx) × (cosx/sinx)

= 1 × 1

= 1

(1 – sin x)/(1 + sin x) = (sec x – tan x)2

Ans – (1 – sin x)/(1 + sin x)

= (1 – sin x)2/(1 – sin x) (1 + sin x),[Multiplying numerator and denominator by (1 – sin x)
= (1 – sin x)2/(1 – sin2 x)

= (1 – sin x)2/(cos2 x), [as we know sin2 θ + cos2 θ = 1 ⇒ cos2 θ = 1 – sin2 θ]

= {(1 – sin x)/cos x}2

= (1/cos x – sin x/cos x)2

= (sec x – tan x)2 

Conclusion

The fundamental six trigonometric functions have a domain input value that is an angle of a right triangle and a numeric answer that is the range. The domain of the trigonometric function f(x) = sinθ  (commonly known as the ‘trig function’) θ is the angle represented in degrees or radians, and the range is [-1, 1]. Similarly, all other functions have a domain and range. Calculus, geometry, and algebra all make heavy use of trigonometric functions.