Trigonometric Ratios and Quadrants

Finding Trigonometric Ratios

• Given sin θ = 2/3 and tan θ > 0, we need to determine the other trigonometric ratios.

Determining the Quadrant

• Since sin θ is positive, θ lies in either the first or second quadrant.
• Since tan θ is positive, θ lies in either the first or third quadrant.
• Combining the conditions, θ lies in the first quadrant.

Calculating cos θ

• Using the Pythagorean identity: sin²θ + cos²θ = 1
• Substituting sin θ = 2/3: (2/3)² + cos²θ = 1
• Simplifying: 4/9 + cos²θ = 1
• Solving for cos²θ: cos²θ = 5/9
• Taking the square root: cos θ = ±√(5/9) = ±√5/3

Determining the Sign of cos θ

• Since θ is in the first quadrant, cos θ is positive.
• Therefore, cos θ = √5/3

Calculating tan θ

• Using the definition of tangent: tan θ = sin θ / cos θ
• Substituting the known values: tan θ = (2/3) / (√5/3)
• Simplifying: tan θ = 2/√5
• Rationalizing the denominator: tan θ = 2√5/5

Calculating csc θ

• Using the reciprocal identity: csc θ = 1/sin θ
• Substituting sin θ = 2/3: csc θ = 1 / (2/3)
• Simplifying: csc θ = 3/2

Calculating sec θ

• Using the reciprocal identity: sec θ = 1/cos θ
• Substituting cos θ = √5/3: sec θ = 1 / (√5/3)
• Simplifying: sec θ = 3/√5 = 3√5/5

Calculating cot θ

• Using the reciprocal identity: cot θ = 1/tan θ
• Substituting tan θ = 2√5/5: cot θ = 1 / (2√5/5)
• Simplifying: cot θ = √5/2

Summary of Results

• sin θ = 2/3
• cos θ = √5/3
• tan θ = 2√5/5
• csc θ = 3/2
• sec θ = 3√5/5
• cot θ = √5/2

This quiz focuses on finding and calculating trigonometric ratios based on given values of sin θ. Students will learn how to determine the quadrant of the angle and use the Pythagorean identity to find other trigonometric functions. It's an essential exercise for mastering trigonometry concepts.