Lesson Notes By Weeks and Term v5 - Grade 10

Lesson Video

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Performance objectives

• Define trig ratios for angles from 0° to 360° on the Cartesian plane.
• Determine the quadrant of an angle from the signs of its trig ratios.
• Use reduction formulae to simplify expressions.
• Solve basic trigonometric equations.

Lesson summary

Trigonometry, derived from Greek words meaning "triangle measurement," is a branch of mathematics that studies relationships involving lengths and angles of triangles. It extends to many fields, including engineering, navigation, physics, and even music. This week, we are specifically focusing on extending our understanding of trigonometric ratios to any angle, not just acute angles within right-angled triangles. This is crucial because real-world problems rarely confine themselves to angles between 0° and 90°. Imagine civil engineers planning a bridge support – they need to calculate angles far beyond acute. Or think about satellite communication, where signals bounce off at various angles.

Lesson notes

2.1 The Cartesian Plane and Trigonometric Ratios We extend our understanding of trigonometric ratios by placing angles in the Cartesian plane (x-y plane). An angle, θ, is in standard position if its vertex is at the origin (0,0) and its initial arm lies along the positive x-axis. The terminal arm is where the angle "stops." Any point (x, y) on the terminal arm can be used to define the trigonometric ratios. Let r be the distance from the origin to the point (x, y) on the terminal arm (i.e., r = √(x² + y²)).

Then: Sine (sin θ) = y/r Cosine (cos θ) = x/r Tangent (tan θ) = y/x Note that r is always positive (since it's a distance), but x and y can be positive or negative depending on the quadrant in which the terminal arm lies. 2.2 Quadrants and Signs of Trigonometric Ratios The Cartesian plane is divided into four quadrants. The signs of x and y in each quadrant determine the signs of the trigonometric ratios: *Quadrant I (0° 0, y >

0. All trigonometric ratios are positive. (All) *Quadrant II (90°

0. Sine is positive; cosine and tangent are negative. (Sin) *Quadrant III (180° 0, y <

0. Cosine is positive; sine and tangent are negative. (Cos) A helpful mnemonic for remembering which ratios are positive in each quadrant is "All Students Take Coffee" (ASTC), starting from Quadrant I and moving counter-clockwise. 2.3 Reduction Formulae Reduction formulae allow us to express trigonometric ratios of angles greater than 90° in terms of trigonometric ratios of acute angles (angles between 0° and 90°). This simplifies calculations. Remember to always check the sign based on the quadrant! Quadrant II (90° < θ < 180°): sin (180° - θ) = sin θ cos (180° - θ) = -cos θ tan (180° - θ) = -tan θ Quadrant III (180° < θ < 270°): sin (180° + θ) = -sin θ cos (180° + θ) = -cos θ tan (180° + θ) = tan θ Quadrant IV (270° < θ < 360°): sin (360° - θ) = -sin θ cos (360° - θ) = cos θ tan (360° - θ) = -tan θ You can also use reduction formulas with 90° ± θ and 270° ± θ but that changes the trigonometric function. This is not covered extensively at Grade 10 level and is better addressed in later grades. It involves the concept of co-functions (sin and cos, tan and cot). 2.4 Solving Basic Trigonometric Equations Solving trigonometric equations involves finding the angles that satisfy a given equation.

Here's the general approach: Isolate the trigonometric ratio (e.g., sin θ = 0.5). Determine the reference angle (the acute angle whose sine, cosine, or tangent is equal to the absolute value of the isolated ratio). Use your calculator (sin⁻¹, cos⁻¹, tan⁻¹). Identify the quadrants where the solution(s) lie, based on the sign of the trigonometric ratio. Use the reduction formulae or knowledge of quadrant angles (0°, 90°, 180°, 270°, 360°) to find all solutions within the given interval (usually 0° to 360°).

Worked example

Example 1: Determine the value of sin 210° without using a calculator.

Solution: 210° lies in Quadrant III (180° < 210° < 270°).

We can write 210° as 180° + 30°.

Using the reduction formula, sin (180° + θ) = -sin θ, we have sin 210° = sin (180° + 30°) = -sin 30°.

We know sin 30° = 1/

2. Therefore, sin 210° = -1/

2. Example 2: Determine the value of cos 300° without using a calculator.

Solution: 300° lies in Quadrant IV (270° < 300° < 360°).

We can write 300° as 360° - 60°.

Using the reduction formula, cos (360° - θ) = cos θ, we have cos 300° = cos (360° - 60°) = cos 60°.

We know cos 60° = 1/

2. Therefore, cos 300° = 1/

2. Example 3: Solve the equation 2 sin θ = 1 for 0° ≤ θ ≤ 360°.

Solution:

Isolate sin θ: sin θ = 1/

2. Find the reference angle: θ = sin⁻¹(1/2) = 30°.

Determine the quadrants: sin θ is positive in Quadrants I and I

I. Solutions:

Quadrant I: θ = 30°

Quadrant II: θ = 180° - 30° = 150°

Therefore, the solutions are θ = 30° and θ = 150°.

Example 4: If the point P(-3, 4) lies on the terminal arm of angle θ, determine sin θ, cos θ, and tan θ.

Solution:

Find r: r = √((-3)² + 4²) = √(9 + 16) = √25 = 5

sin θ = y/r = 4/5

cos θ = x/r = -3/5

tan θ = y/x = 4/-3 = -4/3

Guided Practice (With Solutions)