USTET Mathematics — TrigonometryStudy Notes

The unit circle is the foundation of trigonometry. It's a circle centered at the origin with radius 1, described by the equation x² + y² = 1. **Converting Between Degrees and Radians:** The conversion formula is: Radians = Degrees × (π/180) **Step-by-Step Example 1:** Convert 135° to radians Step 1: Write the conversion formula: radians = degrees × (π/180) Step 2: Substitute 135°: radians = 135 × (π/180) Step 3: Simplify: radians = 135π/180 = 3π/4 Step 4: Answer: 135° = 3π/4 radians **Step-by-Step Example 2:** Convert 17π/12 radians to degrees Step 1: Write the conversion formula: degrees = radians × (180/π) Step 2: Substitute 17π/12: degrees = (17π/12) × (180/π) Step 3: Simplify: degrees = (17 × 180)/12 = 3060/12 = 255° Step 4: Answer: 17π/12 radians = 255° **Arc Length Formula:** - When angle θ is in degrees: a = πrθ/180° - When angle θ is in radians: a = rθ **Common Mistake to Avoid:** Always check if the angle is in degrees or radians before applying formulas. Many students forget to convert and get wrong answers.

The six trigonometric functions relate the sides of right triangles to their angles. **Primary Functions (SOH-CAH-TOA):** - sin θ = Opposite/Hypotenuse (SOH) - cos θ = Adjacent/Hypotenuse (CAH) - tan θ = Opposite/Adjacent (TOA) **Reciprocal Functions (CHO-SHA-CAO):** - csc θ = Hypotenuse/Opposite = 1/sin θ (CHO) - sec θ = Hypotenuse/Adjacent = 1/cos θ (SHA) - cot θ = Adjacent/Opposite = 1/tan θ (CAO) **Step-by-Step Example 1:** Find cos θ for a triangle with adjacent = 18, opposite = 24 Step 1: Identify what we need: cos θ = Adjacent/Hypotenuse Step 2: Find the hypotenuse using Pythagorean theorem: h² = 18² + 24² Step 3: Calculate: h² = 324 + 576 = 900 Step 4: Solve: h = √900 = 30 Step 5: Apply cosine formula: cos θ = 18/30 = 3/5 Step 6: Answer: cos θ = 3/5 **Step-by-Step Example 2:** Car on 30° incline travels 100m, find height Step 1: Identify the triangle: hypotenuse = 100m, angle = 30°, find opposite side Step 2: Choose appropriate function: sin θ = opposite/hypotenuse Step 3: Substitute values: sin 30° = height/100 Step 4: Use special angle: sin 30° = 1/2 Step 5: Solve: 1/2 = height/100, so height = 50m Step 6: Answer: The hill is 50 meters high **Problem-Solving Strategy:** Always draw a diagram and label the sides relative to the given angle before choosing which trigonometric function to use.

Certain angles appear frequently in problems and have exact trigonometric values that you should memorize. **Special Angles: 30°, 45°, 60°** | Function | 30° | 45° | 60° | |----------|-----|-----|-----| | sin θ | 1/2 | √2/2 | √3/2 | | cos θ | √3/2 | √2/2 | 1/2 | | tan θ | √3/3 | 1 | √3 | | csc θ | 2 | √2 | 2√3/3 | | sec θ | 2√3/3 | √2 | 2 | | cot θ | √3 | 1 | √3/3 | **How to Remember These Values:** 1. For sine: 1/2, √2/2, √3/2 (increasing pattern) 2. For cosine: √3/2, √2/2, 1/2 (decreasing pattern) 3. For tangent: use tan θ = sin θ/cos θ **Step-by-Step Example:** Verify that tan 60° = √3 Step 1: Use the ratio identity: tan θ = sin θ/cos θ Step 2: Substitute known values: tan 60° = sin 60°/cos 60° Step 3: Insert special angle values: tan 60° = (√3/2)/(1/2) Step 4: Simplify: tan 60° = (√3/2) × (2/1) = √3 Step 5: Answer: tan 60° = √3 ✓ **Memory Tip:** Create a 30-60-90 triangle with sides 1, √3, 2 and a 45-45-90 triangle with sides 1, 1, √2 to derive these values whenever you forget them.

In the coordinate plane, trigonometric function signs depend on which quadrant the angle terminates in. **Quadrant Sign Rules:** - Quadrant I (0° to 90°): All functions positive - Quadrant II (90° to 180°): Only sin and csc positive - Quadrant III (180° to 270°): Only tan and cot positive - Quadrant IV (270° to 360°): Only cos and sec positive **Memory Device:** "All Students Take Calculus" - All (QI), Students (QII - sin), Take (QIII - tan), Calculus (QIV - cos) **Step-by-Step Example 1:** Find cos 240° Step 1: Determine quadrant: 240° is between 180° and 270°, so it's in QIII Step 2: Find reference angle: 240° - 180° = 60° Step 3: Determine sign: cosine is negative in QIII Step 4: Apply reference angle: cos 240° = -cos 60° Step 5: Use special angle value: cos 240° = -(1/2) = -1/2 Step 6: Answer: cos 240° = -1/2 **Step-by-Step Example 2:** Find angle θ if sin θ = -√2/2 and cos θ > 0 Step 1: Identify reference angle: |sin θ| = √2/2 means reference angle is 45° Step 2: Determine quadrant: sin θ < 0 and cos θ > 0 means QIV Step 3: Find angle in QIV: 360° - 45° = 315° Step 4: Verify: sin 315° = -√2/2 ✓, cos 315° = √2/2 > 0 ✓ Step 5: Answer: θ = 315° **Problem-Solving Strategy:** Always determine the quadrant first, then find the reference angle, then apply the appropriate sign.

Trigonometric identities are equations that are true for all valid values of the variables. They're essential for solving complex trigonometric problems. **Fundamental Identities:** 1. **Reciprocal:** csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ 2. **Quotient:** tan θ = sin θ/cos θ, cot θ = cos θ/sin θ 3. **Pythagorean:** sin² θ + cos² θ = 1, 1 + tan² θ = sec² θ, 1 + cot² θ = csc² θ **Step-by-Step Example 1:** Find sec θ if sin θ = 1/3 Step 1: Use Pythagorean identity: sin² θ + cos² θ = 1 Step 2: Substitute given value: (1/3)² + cos² θ = 1 Step 3: Solve for cos² θ: 1/9 + cos² θ = 1 Step 4: Isolate cos² θ: cos² θ = 1 - 1/9 = 8/9 Step 5: Find cos θ: cos θ = ±√(8/9) = ±2√2/3 Step 6: Find sec θ: sec θ = 1/cos θ = ±3/(2√2) = ±3√2/4 Step 7: Answer: sec θ = ±3√2/4 (sign depends on quadrant) **Step-by-Step Example 2:** Verify the identity: tan² θ + 1 = sec² θ Step 1: Start with left side: tan² θ + 1 Step 2: Use quotient identity: (sin θ/cos θ)² + 1 Step 3: Simplify: sin² θ/cos² θ + 1 Step 4: Common denominator: sin² θ/cos² θ + cos² θ/cos² θ Step 5: Combine fractions: (sin² θ + cos² θ)/cos² θ Step 6: Use Pythagorean identity: 1/cos² θ Step 7: Apply reciprocal identity: sec² θ Step 8: Answer: Identity verified ✓ **Problem-Solving Strategy:** When working with identities, start with the more complex side and use known identities to simplify toward the simpler side.

These laws extend trigonometry beyond right triangles to solve any triangle when given sufficient information. **Law of Sines:** sin α/a = sin β/b = sin γ/c Use when you have: angle-side-angle (ASA), angle-angle-side (AAS), or side-side-angle (SSA) **Law of Cosines:** - a² = b² + c² - 2bc cos α - b² = a² + c² - 2ac cos β - c² = a² + b² - 2ab cos γ Use when you have: side-side-side (SSS) or side-angle-side (SAS) **Step-by-Step Example 1:** Triangle with sides a = 8, b = 10, and angle C = 60°. Find side c. Step 1: Identify given information: two sides and included angle (SAS) Step 2: Choose Law of Cosines: c² = a² + b² - 2ab cos C Step 3: Substitute values: c² = 8² + 10² - 2(8)(10)cos 60° Step 4: Use special angle: c² = 64 + 100 - 160(1/2) Step 5: Calculate: c² = 164 - 80 = 84 Step 6: Solve: c = √84 = 2√21 Step 7: Answer: c = 2√21 ≈ 9.17 **Step-by-Step Example 2:** Triangle with sides a = 5, b = 7, c = 10. Find angle A. Step 1: Identify given information: three sides (SSS) Step 2: Use Law of Cosines: a² = b² + c² - 2bc cos A Step 3: Rearrange for cos A: cos A = (b² + c² - a²)/(2bc) Step 4: Substitute: cos A = (7² + 10² - 5²)/(2 × 7 × 10) Step 5: Calculate: cos A = (49 + 100 - 25)/140 = 124/140 = 31/35 Step 6: Find angle: A = cos⁻¹(31/35) ≈ 27.97° Step 7: Answer: A ≈ 28° **When to Use Which Law:** - Use Law of Sines for ASA, AAS, or SSA cases - Use Law of Cosines for SSS or SAS cases - Law of Cosines reduces to Pythagorean theorem when angle = 90°