Question

Find the value of theta
$\bf{\: if \sin( \theta + 36) = \cos \: ( \theta) where \: \theta \: is \: acute \: angle}$
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Answer 1

Given: sin(θ + 36°) = cos(θ), where θ is acute angle.

To find: The value of theta (θ).

Solution:

Consider the equation;

⇒ sin(θ + 36°) = cos(θ)

⇒ sin(θ + 36°) = sin(90° - θ) [∵ cos(θ) = sin(90° - θ)]

⇒ θ + 36° = 90° - θ

⇒ θ + θ = 90° - 36°

⇒ 2θ = 54°

⇒ θ = 54°/2

⇒ θ = 27°

Hence, the value of theta is 27°.

Additional Information:

1. Relationship between sides and T-Ratios.

• sin(θ) = Height/Hypotenuse
• cos(θ) = Base/Hypotenuse
• tan(θ) = Height/Base
• cot(θ) = Base/Height
• sec(θ) = Hypotenuse/Base
• cosec(θ) = Hypotenuse/Height

2. Square formulae.

• sin²(θ) + cos²(θ) = 1
• 1 + tan²(θ) = sec²(θ)
• 1 + cot²(θ) = cosec²(θ)

3. Reciprocal Relationship.

• sin(θ) = 1/cosec(θ)
• cos(θ) = 1/sec(θ)
• tan(θ) = 1/cot(θ)
• cot(θ) = 1/tan(θ)
• sin(θ)/cos(θ) = 1/cot(θ)
• cos(θ)/sin(θ) = 1/tan(θ)

4. Trigonometric functions of multiple of angles.

• sin2(θ) = 2sin(θ)cos(θ)
• cos2(θ) = cos²(θ) - sin²(θ)
• cos2(θ) = 2cos²(θ) - 1
• cos2(θ) = 1 - sin²(θ)
• tan(θ) = 2tan(θ)/1 - tan²(θ)

5. Sign of Trigonometric ratios in Quadrants.

• sin (90° - θ) = cos(θ)
• cos (90° - θ) = sin(θ)
• tan (90° - θ) = cot(θ)
• csc (90° - θ) = sec(θ)
• sec (90° - θ) = csc(θ)
• cot (90° - θ) = tan(θ)
• sin (90° + θ) = cos(θ)
• cos (90° + θ) = -sin(θ)
• tan (90° + θ) = -cot(θ)
• csc (90° + θ) = sec(θ)
• sec (90° + θ) = -csc(θ)
• cot (90° + θ) = -tan(θ)
• sin (180° - θ) = sin(θ)
• cos (180° - θ) = -cos(θ)
• tan (180° - θ) = -tan(θ)
• csc (180° - θ) = csc(θ)
• sec (180° - θ) = -sec(θ)
• cot (180° - θ) = -cot(θ)
• sin (180° + θ) = -sin(θ)
• cos (180° + θ) = -cos(θ)
• tan (180° + θ) = tan(θ)
• csc (180° + θ) = -csc(θ)
• sec (180° + θ) = -sec(θ)
• cot (180° + θ) = cot(θ)
• sin (270° - θ) = -cos(θ)
• cos (270° - θ) = -sin(θ)
• tan (270° - θ) = cot(θ)
• csc (270° - θ) = -sec(θ)
• sec (270° - θ) = -csc(θ)
• cot (270° - θ) = tan(θ)
• sin (270° + θ) = -cos(θ)
• cos (270° + θ) = sin(θ)
• tan (270° + θ) = -cot(θ)
• csc (270° + θ) = -sec(θ)
• sec (270° + θ) = cos(θ)
• cot (270° + θ) = -tan(θ)

Answer 2

PROVIDED INFORMATION :-

if sin ( θ +36) = cos (θ) where is acute angle

QUESTION :-

Find the value of theta if sin ( θ +36) = cos (θ) where is acute angle

TO FIND :-

Find the value of theta = ?

SOLUTION :-

Find the condition :-

We have that,

sec θ . sin ( 36° + θ ) = 1

sin (36° +θ ) = cos θ

sin (36° +θ ) = sin (90° - θ )

(36° + θ ) = (90° - θ )

2θ = 54°

θ = 54 /2

θ = 27

The value of θ = 27°

ADDITIONAL INFORMATION :-

An angle that measures less than is called an acute angle.

An angle smaller than a right angle is less than is called an acute angle.