Pythagorean Identity

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Pythagorean Identity · Grade 11 · Worksheet 2

1. A unit circle is drawn on a coordinate plane with center at the origin. Point P on the circle has coordinates (0.6, 0.8). A vertical line segment is drawn from P down to the x-axis, and a horizontal line segment is drawn from P to the y-axis, forming a right triangle with the axes. Using the definitions of sine and cosine from the unit circle and the geometric relationship between the sides of this triangle, prove algebraically that sin²θ + cos²θ = 1. Answer: 1 Solution: Identify the coordinates of point P on the unit circle. The point is P = (0.6, 0.8). The center is at (0,0) and the radius is 1.
   Full step-by-step solution

   Step 1: Identify the coordinates of point P on the unit circle. The point is P = (0.6, 0.8). The center is at (0,0) and the radius is 1. Step 2: In the unit circle definition, for an angle θ measured from the positive x-axis to the radius OP, the x-coordinate of P equals cos θ and the y-coordinate equals sin θ. So cos θ = 0.6 and sin θ = 0.8. Step 3: The vertical line from P to the x-axis has length equal to the y-coordinate, which is 0.8. The horizontal line from P to the y-axis has length equal to the x-coordinate, which is 0.6. Together with the radius (hypotenuse = 1), these form a right triangle. Step 4: Apply the Pythagorean theorem to this right triangle: (horizontal leg)² + (vertical leg)² = (hypotenuse)². (0.6)² + (0.8)² = 0.36 + 0.64 = 1.00 = 1². Step 5: Replace the leg lengths with sin θ and cos θ: cos²θ + sin²θ = 1. Step 6: Therefore, sin²θ + cos²θ = 1 is proven by the geometry of the unit circle. The answer is 1.

2. Mason draws a unit circle on a coordinate plane centered at the origin. From the point where the circle intersects the positive x-axis at (1, 0), he travels counterclockwise along the circumference to a point P such that the arc length from (1, 0) to P is 9 units. Using the unit circle definitions of sine and cosine, prove algebraically that sin²θ + cos²θ = 1, where θ is the angle subtended by the arc from the positive x-axis to point P. Answer: 1 Solution: Recall the unit circle definition. A unit circle has radius r = 1 and is centered at (0, 0). The equation of this circle is x² + y² = 1.
   Full step-by-step solution

   Step 1: Recall the unit circle definition. A unit circle has radius r = 1 and is centered at (0, 0). The equation of this circle is x² + y² = 1. Step 2: On the unit circle, for an angle θ measured from the positive x-axis counterclockwise, the coordinates of point P are (cos θ, sin θ). Step 3: Since point P lies on the unit circle, its coordinates must satisfy the circle's equation: x² + y² = 1. Step 4: Substitute x = cos θ and y = sin θ into the circle equation: (cos θ)² + (sin θ)² = 1 Step 5: This is exactly the Pythagorean identity: cos²θ + sin²θ = 1, which can be rewritten as sin²θ + cos²θ = 1. Step 6: The arc length of 9 units simply determines a specific angle θ (since arc length = r × θ = 1 × θ = 9 radians), but the proof holds for any angle θ on the unit circle. Therefore, sin²θ + cos²θ = 1 has been proven algebraically using the unit circle. The answer is 1.

3. Liam is designing a triangular support structure for a new bridge. He knows that for a particular angle θ in the triangle, the ratio of the opposite side to the hypotenuse is represented by sin(θ). Using the geometric relationships within a right triangle inscribed in a unit circle, prove algebraically that for any angle θ, the square of this sine ratio plus the square of the cosine ratio must always equal 1. Answer: sin²θ + cos²θ = 1 Solution: Draw a unit circle. A unit circle is a circle with radius 1, centered at the origin (0,0) of a coordinate plane. Place a right triangle inside the circle.
   Full step-by-step solution

   Let's prove that sin²θ + cos²θ = 1 using a right triangle in the unit circle. Step 1: Draw a unit circle. A unit circle is a circle with radius 1, centered at the origin (0,0) of a coordinate plane. Step 2: Place a right triangle inside the circle. Imagine an angle θ measured from the positive x-axis. Draw a line from the center of the circle to a point on the circle. Let's call this point P. The coordinates of point P are (x, y). Step 3: Identify the triangle's sides. From point P, drop a perpendicular line down to the x-axis. This creates a right triangle with: - The hypotenuse as the line from the origin (0,0) to point P (x,y). Since the circle has radius 1, the hypotenuse length is 1. - The horizontal side is along the x-axis from (0,0) to (x,0). Its length is x. - The vertical side is from (x,0) to (x,y). Its length is y. Step 4: Define sine and cosine using the triangle. In this right triangle: - sin(θ) = (opposite side) / (hypotenuse) = y / 1 = y - cos(θ) = (adjacent side) / (hypotenuse) = x / 1 = x So we have established that for any angle θ on the unit circle: cos(θ) = x sin(θ) = y Step 5: Use the equation of the unit circle. The equation for a circle centered at the origin with radius 1 is: x² + y² = 1 Step 6: Substitute the trigonometric identities. Since we found that x = cos(θ) and y = sin(θ), we can substitute these into the circle equation. Replace x with cos(θ) and y with sin(θ): (cos(θ))² + (sin(θ))² = 1 Step 7: Write using standard notation. (cos(θ))² is written as cos²θ (sin(θ))² is written as sin²θ Therefore: sin²θ + cos²θ = 1 Conclusion: We have shown algebraically that for any angle θ, the square of the sine ratio plus the square of the cosine ratio always equals 1. This is known as the Pythagorean Identity.

4. Given sin θ = 5/13, find cos θ using sin²θ + cos²θ = 1 Answer: 12/13 Solution: Start with the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: (5/13)² + cos²θ = 1 Calculate (5/13)² = 25/169 Write the equation: 25/169 + cos²θ = 1 Subtract 25/169 from both sides: cos²θ = 1 - 25/169 Calculate 1 - 25/169 = 169/169 - 25/169 = 144/169 Take the square root: cos…
   Full step-by-step solution

   Step 1: Start with the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: (5/13)² + cos²θ = 1 Step 3: Calculate (5/13)² = 25/169 Step 4: Write the equation: 25/169 + cos²θ = 1 Step 5: Subtract 25/169 from both sides: cos²θ = 1 - 25/169 Step 6: Calculate 1 - 25/169 = 169/169 - 25/169 = 144/169 Step 7: Take the square root: cos θ = ±√(144/169) = ±12/13 Step 8: Since the problem doesn't specify a quadrant, we take the positive value: cos θ = 12/13 Final answer: 12/13

5. Given sin θ = 12/13, find cos θ using sin²θ + cos²θ = 1 Answer: 5/13 Solution: Write down the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: (12/13)² + cos²θ = 1 Calculate (12/13)² = 144/169 Rewrite the equation: 144/169 + cos²θ = 1 Subtract 144/169 from both sides: cos²θ = 1 - 144/169 Convert 1 to 169/169: cos²θ = 169/169 - 144/169 Simplify: cos²θ =…
   Full step-by-step solution

   Step 1: Write down the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: (12/13)² + cos²θ = 1 Step 3: Calculate (12/13)² = 144/169 Step 4: Rewrite the equation: 144/169 + cos²θ = 1 Step 5: Subtract 144/169 from both sides: cos²θ = 1 - 144/169 Step 6: Convert 1 to 169/169: cos²θ = 169/169 - 144/169 Step 7: Simplify: cos²θ = 25/169 Step 8: Take the square root of both sides: cos θ = ±5/13 Step 9: Since no quadrant is specified, we take the positive value as the principal answer: cos θ = 5/13

6. Liam is designing a triangular support bracket for a construction project. He knows that for a right triangle with angle θ, the relationship between the opposite side and hypotenuse is given by sin(θ), and between the adjacent side and hypotenuse is given by cos(θ). If he represents the sides using these trigonometric functions, what fundamental identity must hold true for all possible angles θ in his design? Answer: sin²θ + cos²θ = 1 Solution: Liam is working with a right triangle. - sin(θ) = opposite / hypotenuse - cos(θ) = adjacent / hypotenuse - Hypotenuse = \( h \) - Opposite side = \( o \) - Adjacent side = \( a \) sin(θ) = \( o / h \) cos(θ) = \( a / h \) \( o^2 + a^2 = h^2 \) Substitute sin(θ) and cos(θ) into the Pythagorean…
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understanding the problem** Liam is working with a right triangle. In a right triangle, for a given angle θ: - sin(θ) = opposite / hypotenuse - cos(θ) = adjacent / hypotenuse Let’s call: - Hypotenuse = \( h \) - Opposite side = \( o \) - Adjacent side = \( a \) Then: sin(θ) = \( o / h \) cos(θ) = \( a / h \) --- **Step 2: Relating sides using Pythagorean theorem** For any right triangle: \( o^2 + a^2 = h^2 \) --- **Step 3: Substitute sin(θ) and cos(θ) into the Pythagorean equation** From sin(θ) = \( o / h \), we get \( o = h \cdot \sin(θ) \) From cos(θ) = \( a / h \), we get \( a = h \cdot \cos(θ) \) Substitute into \( o^2 + a^2 = h^2 \): \( (h \cdot \sin(θ))^2 + (h \cdot \cos(θ))^2 = h^2 \) --- **Step 4: Simplify the equation** \( h^2 \sin^2(θ) + h^2 \cos^2(θ) = h^2 \) Factor \( h^2 \) on the left: \( h^2 (\sin^2(θ) + \cos^2(θ)) = h^2 \) --- **Step 5: Cancel \( h^2 \) (since \( h > 0 \))** We get: \( \sin^2(θ) + \cos^2(θ) = 1 \) --- **Step 6: Conclusion** This identity holds for all possible angles θ in a right triangle (and in fact for all real θ in trigonometry), so it must be true in Liam’s design. --- **Final answer:** sin²θ + cos²θ = 1

7. Given cos θ = 15/17 and θ is in quadrant IV, find sin θ using sin²θ + cos²θ = 1 Answer: -8/17 Solution: Start with the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: sin²θ + (15/17)² = 1 Calculate (15/17)² = 225/289 Rewrite the equation: sin²θ + 225/289 = 1 Subtract 225/289 from both sides: sin²θ = 1 - 225/289 Convert 1 to 289/289: sin²θ = 289/289 - 225/289 Simplify: sin²θ =…
   Full step-by-step solution

   Step 1: Start with the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: sin²θ + (15/17)² = 1 Step 3: Calculate (15/17)² = 225/289 Step 4: Rewrite the equation: sin²θ + 225/289 = 1 Step 5: Subtract 225/289 from both sides: sin²θ = 1 - 225/289 Step 6: Convert 1 to 289/289: sin²θ = 289/289 - 225/289 Step 7: Simplify: sin²θ = 64/289 Step 8: Take the square root: sin θ = ±√(64/289) = ±8/17 Step 9: Since θ is in quadrant IV where sine is negative, sin θ = -8/17 Final answer: -8/17