Step by Step
1
The core identity
sin²θ + cos²θ = 1, true for every angle θ. This comes directly from the Pythagorean theorem applied to the unit circle, where the hypotenuse is always 1.
2
Rearranging for either term
You can isolate either term: sin²θ = 1 − cos²θ, or cos²θ = 1 − sin²θ — useful whenever you know one value and need the other.
3
Deriving the tangent-secant identity
Dividing the entire original identity by cos²θ gives: tan²θ + 1 = sec²θ.
4
Deriving the cotangent-cosecant identity
Dividing the entire original identity by sin²θ instead gives: 1 + cot²θ = csc²θ.
Applied Walkthrough
1
Given that sin(θ) = 3/5 and θ is in Quadrant I, find cos(θ) using the Pythagorean identity.
2
Rearrange the identity: cos²θ = 1 − sin²θ = 1 − (3/5)² = 1 − 9/25 = 16/25.
3
Take the square root: cos(θ) = ±4/5. Since θ is in Quadrant I (where cosine is positive, per ASTC), cos(θ) = 4/5.
4
This same triangle (3-4-5) is a classic Pythagorean triple, confirming the result geometrically: a right triangle with legs 3 and 4 has a hypotenuse of 5.
Exam Application
Exams test whether you can rearrange the Pythagorean identity to solve for a missing trig value, correctly apply the quadrant's sign (using ASTC) when taking a square root, and recognize the two derived identities (tan²θ+1=sec²θ and 1+cot²θ=csc²θ).
⚠ Common Trap
The most common trap is forgetting to apply the correct sign after taking a square root — the identity itself only gives you the squared value, so you must use the quadrant information (via ASTC) to determine whether the actual value is positive or negative.
✓ Quick Self-Check
1. What is the Pythagorean identity?
sin²θ + cos²θ = 1.
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2. What identity do you get by dividing the Pythagorean identity by cos²θ?
tan²θ + 1 = sec²θ.
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3. What identity do you get by dividing the Pythagorean identity by sin²θ?
1 + cot²θ = csc²θ.
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4. If sin(θ) = 3/5 and θ is in Quadrant I, what is cos(θ)?
4/5.
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5. Why must you check the quadrant after taking a square root in these problems?
Because the identity only gives you the squared value, so the quadrant determines whether the actual (unsquared) value is positive or negative.
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