Why sin²θ + cos²θ = 1 Always Holds True? Where Does sin²θ + cos²θ = 1 Come From?
Trigonometry is full of magical formulas, but one identity stands out as the most famous and the most important:
sin²θ + cos²θ = 1
This is called the Pythagorean Identity, and it is the backbone of countless trigonometric simplifications, exam questions, and real-life applications in physics, engineering, and navigation.
The identity \[ \sin^2 \theta + \cos^2 \theta = 1 \] is a fundamental truth in trigonometry and holds for all real angles \( \theta \).sin²θ + cos²θ = 1
1) Right-Triangle Definition - Based on Right Triangle Geometry (Pythagoras Theorem)
By definition of trigonometric ratios::
\( \sin \theta = \frac{a}{c} \)
\( \cos \theta = \frac{b}{c} \)
Now square both:
\(\sin^2\theta = \frac{a^2}{c^2}\), \(cos^2\theta = \frac{b^2}{c^2}\)
Add them:
\[ \sin^2 \theta + \cos^2 \theta = \frac{a^2 + b^2}{c^2}. \]
By the Pythagorean Theorem,
\( a^2 + b^2 = c^2 \),
\[ \sin^2 \theta + \cos^2 \theta= \frac{a^2 + b^2}{c^2}= \frac{c^2}{c^2} = 1. \]
2. Based on Unit Circle - Unit Circle Explanation
3) Why It Always Holds and Why It’s Called an Identity
It follows directly from the Pythagorean Theorem via triangle definitions of sine and cosine.It follows from the unit circle equation via coordinate geometry.
Both viewpoints are consistent, so the identity is universally valid for all angles \( \theta \).
Why Is This Identity Important?
This is the most fundamental relation in trigonometry. From this identity, other useful identities are derived:
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
It helps simplify complex trigonometric equations.
It connects trigonometry with geometry.
It is used in calculus, physics, navigation, and astronomy.
Solved Examples
Example 1:
If cos θ = 5/13, find sin θ.
cos²θ + sin²θ = 1
(5/13)² + sin²θ = 1
25/169 + sin²θ = 1
sin²θ = 1 – 25/169
sin²θ = 144/169
sin θ = 12/13
Example 2:
Simplify sin²30° + cos²30°.
sin 30° = 1/2,cos 30° = √3/2
So,
sin²30° + cos²30° = 1/4 + 3/4 = 1
Frequently Asked Questions (FAQs) on sin²θ + cos²θ = 1
Q1. Why is sin²θ + cos²θ = 1 called an identity?
Because it is true for all values of θ, unlike an equation which may hold only for specific values.
Q2. Can sin²θ + cos²θ = 1 be used in physics?
Yes,
It is used in wave motion, oscillations, and alternating current equations.
Q3. Does this identity work for negative angles?
Yes,
because squaring removes the sign.
Q4. Is this related to vectors?
Yes,
when a vector is resolved into x and y components using cos θ and sin θ, the total magnitude is always preserved using this identity.
Q5. Can it be extended to 3D?
Yes,
in 3D geometry:
cos²α + cos²β + cos²γ = 1 (direction cosines).
Q6. What is the graphical meaning?
It represents a circle of radius 1 on the coordinate plane.
Q7. What common mistakes do students make?
- Confusing sin²θ with (sin θ)².
- Thinking sin²θ + cos²θ = (sin θ + cos θ)² (❌ wrong).
- Forgetting it is always equal to 1.
Q8. Who discovered sin²θ + cos²θ = 1?
This identity is directly linked to the Pythagoras Theorem and was formalized as trigonometry developed through the work of Greek, and Indian mathematicians.Q9. Does sin²θ + cos²θ = 1 hold true for all angles?
Yes,It is valid for all real values of θ (acute, obtuse, reflex, negative angles, and even beyond 360°). It also holds when angles are measured in radians.
Q10. Can this identity be used in calculus?
Absolutely. In differentiation and integration, this identity helps simplify expressions like ∫ sin²θ dθ or ∫ cos²θ dθ.Q11. What are the derived identities from sin²θ + cos²θ = 1?
By dividing both sides by cos²θ, we get:1 + tan²θ = sec²θ.
By dividing both sides by sin²θ, we get:
1 + cot²θ = csc²θ.
Q12. Why is this called a Pythagorean Identity?
Because it directly comes from the Pythagoras Theorem applied to a right triangle (the relationship between the square of the sides produces this identity).Q13. How can I quickly remember this identity?
Think: “Circle + Triangle = 1”- Circle → x² + y² = 1
- Triangle → (sin)² + (cos)² = 1
Q14. Where do we see this formula in physics?
Common places:- Wave motion (sine & cosine waves)
- Circular motion (velocity components)
- Electrical engineering (AC power equations)
Q15. Does sin²θ + cos²θ = 1 work in 3D geometry?
Yes, but in 3D it expands into vector forms. In higher dimensions, similar identities are used in physics and computer graphics to preserve lengths and normalize vectors.Q16. Is sin²θ + cos²θ = 1 used in competitive exams in India?
Yes, It appears in many exams (SAT, ACT, JEE, GRE, NDA, UPSC, SSC, GMAT, CAT, etc.). A large share of trigonometry problems use this identity directly or indirectly.Q17. Why is this formula so famous compared to other trig formulas?
Because it is the base identity. Many other identities (like 1 + tan²θ = sec²θ) are derived from it. It underpins much of trigonometry.Q18. Can sin²θ + cos²θ = 1 ever be greater or less than 1?
No, The sum is always exactly 1 for any real θ. That invariant value is one of the identity’s defining properties.Q19. How is this identity used in computer graphics?
When objects rotate, positions are calculated using sine and cosine. The identity ensures vector normalization and keeps rotations stable and realistic in games and animations.Q20. Can we prove this using Euler’s formula (e^iθ)?
Yes,Euler’s formula: e^{iθ} = cos θ + i sin θ.
Taking modulus gives
Q21. What is the difference between sin²θ + cos²θ = 1 and sin θ + cos θ = 1?
sin²θ + cos²θ = 1 → always true for all θ.sin θ + cos θ = 1 → not always true; holds only for specific angles (e.g., θ = 0° is one trivial case)
Q22. How does this formula connect with the unit circle?
On the unit circle (radius = 1):x = cosθ, y = sinθ.
So x² + y² = 1 becomes sin²θ + cos²θ = 1.
Practice Questions on sin²θ + cos²θ = 1
Q1. Prove that 1 – sin²θ = cos²θ.
sin²θ + cos²θ = 1
cos²θ = 1 – sin²θ
Q2. If sin θ = 4/5, find cos θ.
sin²θ + cos²θ = 1
(4/5)² + cos²θ = 1
16/25 + cos²θ = 1
cos²θ = 9/25
cos θ = 3/5
Q3. Show that (1 + tan²θ) = sec²θ.
Divide this identity by cos²θ:
(sin²θ/cos²θ) + (cos²θ/cos²θ) = 1/cos²θ
tan²θ + 1 = sec²θ
Q4. Simplify : (cos²θ – sin²θ)² + (2sinθcosθ)².
LHS = cos⁴θ – 2sin²θcos²θ + sin⁴θ + 4sin²θcos²θ
= cos⁴θ + sin⁴θ + 2sin²θcos²θ
= (cos²θ + sin²θ)² = 1
Q5. Simplify sin⁴θ + cos⁴θ.
sin⁴θ + cos⁴θ
= (sin²θ + cos²θ)² – 2sin²θcos²θ
= 1 – 2sin²θcos²θ
The identity sin²θ + cos²θ = 1 is the foundation of trigonometry. It comes directly from the Pythagoras theorem and the unit circle, and it connects mathematics with physics, engineering, and real life. By mastering it, you unlock a powerful tool for solving equations, simplifying expressions, and understanding the world around you.
Now try solving the practice questions and share this article with your friends preparing for exams — because this identity will never leave you in maths.
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