Subscribe to our ▶️ YouTube channel 🔴 for the latest videos, updates, and tips.
Home | About Us | Contact Us | Privacy | Math Blog
Problems on finding the unknown angle using trigonometric identities.
1. Solve: tan θ + cot θ = 2, where 0° < θ < 90°.
Solution:
Here, tan θ + cot θ = 2
⟹ tan θ + 1tanθ = 2
⟹ tan2θ+1tanθ = 2
⟹ tan2 θ + 1 = 2 tan θ
⟹ tan2 θ - 2 tan θ + 1 = 0
⟹ (tan θ - 1)2 = 0
⟹ tan θ – 1 = 0
⟹ tan θ = 1
⟹ tan θ = tan 45°
⟹ θ = 45°.
Therefore, θ = 45°.
2. Is sinθ1–cosθ + sinθ1+cosθ = 4 an identity? If not, find θ (0° < θ < 90°).
Solution:
Here, LHS = sinθ(1+cosθ)+sinθ(1−cosθ)(1–cosθ)(1+cosθ)
= 2sinθ1–cos2θ
= 2sinθsin2θ, [using trigonometric identities, sin2 θ + cos2 θ = 1]
= 2sinθ
Thus, the given equality becomes 2sinθ = 4.
Now, if the equality holds true for all values of θ then the equality is an identity.
Let us take (arbitrarily) θ = 45°.
So, 2sin45° = 21√2 = 2√2
So, sin θ ≠ 4.
Therefore, the equality is not an identity.
It is an equation. Then, from the equation we have,
2sinθ = 4
⟹ sin θ = 12
⟹ sin θ = sin 30°
Therefore, θ = 30°.
3. If 5 cos θ + 12 sin θ = 13, find sin θ.
Solution:
5 cos θ + 12 sin θ = 13
⟹ 5 cos θ = 13 - 12 sin θ
⟹ (5 cos θ)2 = (13 – 12 sin θ)2
⟹ 25 cos2 θ = 169 - 312 sin θ + 144 sin θ2
⟹ 25(1 - sin2 θ) = 169 - 312 sin θ + 144 sin θ2, [using trigonometric identities, sin2 θ + cos2 θ = 1]
⟹ 25 – 25 sin2 θ = 169 – 312 sin θ + 144 sin θ2,
⟹ 169 sin2 θ – 312 sin θ + 144 = 0
⟹ (13 sin θ – 12)2 = 0
Therefore, 13 sin θ – 12 = 0
⟹ sin θ = 1213.
4. If √3sin θ - cos θ = 0, prove that tan 2θ = 2tanθ1–tan2θ.
Solution:
Here, √3sin θ - cos θ = 0
⟹ sinθcosθ = 1√3
⟹ tan θ = 1√3
⟹ tan θ = tan 30°
⟹ θ = 30°
Therefore, tan 2θ = tan (2 × 30°) = tan 60° = √3
Now, 2tanθ1–tan2θ = 2tan30°1–tan230°
= 2×1√31–(1√3)2
= 2√31–13
= 2√323
= 2√3 × 32
= √3.
Therefore, tan 2θ = 2tanθ1–tan2θ. (proved)
From Finding the Unknown Angle to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
Jul 25, 25 12:21 PM
Jul 25, 25 03:15 AM
Jul 24, 25 03:46 PM
Jul 23, 25 11:37 AM
Jul 20, 25 10:22 AM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.