Definition of sine, cosine, tangent, and cotangent

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MODULE: Trigonometric Functions and Identities

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✖️ 1. Identifying opposite and adjacent sides, and the hypotenuse relative to an acute angle

📐 Identifying Triangle Sides

  • The hypotenuse is always the longest side opposite the right angle.
  • The opposite side is across from the angle you're focusing on.
  • The adjacent side is next to the angle you're focusing on (but not the hypotenuse).
  • These labels change when you switch which acute angle you're measuring from.
  • Always identify your reference angle first before labeling sides.

In a right triangle with angle A at the bottom left, if the vertical side is 3 and horizontal side is 4, then relative to angle A: opposite = 3, adjacent = 4, hypotenuse = 5.

💡 Think: opposite is across, adjacent is attached, hypotenuse is the hero (longest).

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1. Identifying opposite and adjacent sides, and the hypotenuse relative to an acute angle

Identifying Opposite and Adjacent Sides, and the Hypotenuse Relative to an Acute Angle

In a right triangle, the hypotenuse is the longest side opposite the right angle. For any acute angle θ\theta in the triangle, the opposite side is the side across from θ\theta, and the adjacent side is the side that forms θ\theta with the hypotenuse.

Intuition: The labels "opposite" and "adjacent" depend entirely on which acute angle you choose as your reference angle.

Core Rules:

  • The hypotenuse is always the same regardless of which acute angle is chosen.
  • The opposite side to angle θ\theta becomes the adjacent side to the other acute angle.
  • All three sides must satisfy the Pythagorean theorem: a2+b2=c2a^2 + b^2 = c^2 where cc is the hypotenuse.

Consequence: Correctly identifying these sides is essential for defining trigonometric ratios, as each ratio compares specific pairs of sides.

Example: In a right triangle with acute angle θ=30∘\theta = 30^\circ, hypotenuse 10, opposite side 5, the adjacent side is 75=53\sqrt{75} = 5\sqrt{3}.

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A right triangle has two shorter sides (legs) of length 3 and 4. These sides form the right angle. Calculate the length of the hypotenuse.

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✖️ 2. Defining sine, cosine, and tangent as geometric ratios (SOH CAH TOA)

🎯 SOH CAH TOA Ratios

  • Sine = Opposite / Hypotenuse (SOH).
  • Cosine = Adjacent / Hypotenuse (CAH).
  • Tangent = Opposite / Adjacent (TOA).
  • These ratios depend only on the angle size, not the triangle size.
  • All three values are always between 0 and 1 for acute angles (except tangent can exceed 1).

For a 3-4-5 triangle with angle A opposite the side of length 3: sin(A) = 3/5, cos(A) = 4/5, tan(A) = 3/4.

💡 Memory: Some Old Hippie Caught Another Hippie Tripping On Acid.

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2. Defining sine, cosine, and tangent as geometric ratios (SOH CAH TOA)

Defining Sine, Cosine, and Tangent as Geometric Ratios (SOH CAH TOA)

For an acute angle θ\theta in a right triangle, sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.

Intuition: These ratios quantify the relationship between an angle's measure and the triangle's side lengths, remaining constant for a given angle regardless of triangle size.

Core Rules:

  • sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} (SOH)
  • cos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} (CAH)
  • tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} (TOA)
  • All three ratios are dimensionless and positive for acute angles.

Consequence: These definitions enable calculation of unknown sides or angles using only partial triangle information.

Example: If θ=30∘\theta = 30^\circ with hypotenuse 8, then opposite =8sin⁡(30∘)=8⋅0.5=4= 8 \sin(30^\circ) = 8 \cdot 0.5 = 4.

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In a right triangle, the side opposite to an acute angle θ\theta has a length of 7, and the hypotenuse has a length of 25. Which trigonometric ratio is equal to 7/25?

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✖️ 3. Defining the reciprocal functions: cosecant, secant, and cotangent

🔄 Reciprocal Trig Functions

  • Cosecant (csc) = 1 / sin = Hypotenuse / Opposite.
  • Secant (sec) = 1 / cos = Hypotenuse / Adjacent.
  • Cotangent (cot) = 1 / tan = Adjacent / Opposite.
  • These are just flipped versions of sine, cosine, and tangent.
  • Reciprocals are undefined when the original function equals zero.

If sin(A) = 3/5, then csc(A) = 5/3. If tan(A) = 3/4, then cot(A) = 4/3.

💡 Flip the fraction: csc flips sin, sec flips cos, cot flips tan.

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3. Defining the reciprocal functions: cosecant, secant, and cotangent

Defining the Reciprocal Functions: Cosecant, Secant, and Cotangent

Cosecant (csc⁡\csc), secant (sec⁡\sec), and cotangent (cot⁡\cot) are defined as the reciprocals of sine, cosine, and tangent respectively.

Intuition: These functions provide alternative perspectives on the same geometric relationships, often simplifying certain calculations or expressions.

Core Rules:

  • csc⁡(θ)=1sin⁡(θ)=hypotenuseopposite\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{hypotenuse}}{\text{opposite}}
  • sec⁡(θ)=1cos⁡(θ)=hypotenuseadjacent\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent}}
  • cot⁡(θ)=1tan⁡(θ)=adjacentopposite\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adjacent}}{\text{opposite}}
  • These functions are undefined when their denominators equal zero (e.g., csc⁡(0∘)\csc(0^\circ) is undefined).

Consequence: Reciprocal functions are particularly useful in identities and when solving equations where the primary trigonometric function appears in a denominator.

Example: For θ=45∘\theta = 45^\circ, csc⁡(45∘)=1sin⁡(45∘)=122=2\csc(45^\circ) = \frac{1}{\sin(45^\circ)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}.

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In a right triangle, the side adjacent to angle θ\theta has a length of 55, the side opposite to θ\theta has a length of 1212, and the hypotenuse has a length of 1313. What is the value of sec⁡(θ)\sec(\theta)?

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✖️ 4. Special right triangles (30-60-90, 45-45-90) and their exact trigonometric values

⭐ Special Right Triangles

  • 45-45-90 triangle: sides are in ratio 1:1:21 : 1 : \sqrt{2}, so sin⁡(45∘)=cos⁡(45∘)=22\sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2} and tan⁡(45∘)=1\tan(45^\circ) = 1.
  • 30-60-90 triangle: sides are in ratio 1:3:21 : \sqrt{3} : 2 (opposite 30, 60, 90 respectively).
  • For 30 degrees: sin⁡(30∘)=12\sin(30^\circ) = \frac{1}{2}, cos⁡(30∘)=32\cos(30^\circ) = \frac{\sqrt{3}}{2}, tan⁡(30∘)=33\tan(30^\circ) = \frac{\sqrt{3}}{3}.
  • For 60 degrees: sin⁡(60∘)=32\sin(60^\circ) = \frac{\sqrt{3}}{2}, cos⁡(60∘)=12\cos(60^\circ) = \frac{1}{2}, tan⁡(60∘)=3\tan(60^\circ) = \sqrt{3}.
  • Memorize these exact values to solve problems without a calculator.

A 30-60-90 triangle with hypotenuse 10 has sides 5 (opposite 30), 535\sqrt{3} (opposite 60), and 10.

💡 45-45-90 is symmetric; 30-60-90 goes 1-root3-2 from smallest angle up.

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4. Special right triangles (30-60-90, 45-45-90) and their exact trigonometric values

Special Right Triangles (30-60-90, 45-45-90) and Their Exact Trigonometric Values

Special right triangles have side ratios that yield exact trigonometric values without approximation. The 45-45-90 triangle has sides in ratio 1:1:21:1:\sqrt{2}, while the 30-60-90 triangle has sides in ratio 1:3:21:\sqrt{3}:2.

Intuition: These triangles provide benchmark values that appear frequently in geometry and calculus.

Core Rules:

  • 45-45-90: sin⁡(45∘)=cos⁡(45∘)=22\sin(45^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}, tan⁡(45∘)=1\tan(45^\circ) = 1
  • 30-60-90: sin⁡(30∘)=12\sin(30^\circ) = \frac{1}{2}, cos⁡(30∘)=32\cos(30^\circ) = \frac{\sqrt{3}}{2}, tan⁡(30∘)=33\tan(30^\circ) = \frac{\sqrt{3}}{3}
  • 30-60-90: sin⁡(60∘)=32\sin(60^\circ) = \frac{\sqrt{3}}{2}, cos⁡(60∘)=12\cos(60^\circ) = \frac{1}{2}, tan⁡(60∘)=3\tan(60^\circ) = \sqrt{3}

Consequence: Memorizing these values eliminates the need for calculators in many standard problems.

Example: In a 30-60-90 triangle with hypotenuse 6, the side opposite 30∘30^\circ is 6⋅12=36 \cdot \frac{1}{2} = 3.

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In a 30-60-90 triangle, the hypotenuse has a length of 1010. What is the length of the side opposite the 30∘30^\circ angle?

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✖️ 5. Using the Pythagorean theorem to verify side relationships and find missing ratios

✅ Pythagorean Verification

  • The Pythagorean theorem states a2+b2=c2a^2 + b^2 = c^2 where c is the hypotenuse.
  • Use it to find a missing side when two sides are known.
  • After finding sides, calculate any trig ratio using SOH CAH TOA.
  • Check your ratios: sin⁡2(θ)+cos⁡2(θ)\sin^2(\theta) + \cos^2(\theta) must always equal 1.
  • This identity comes directly from the Pythagorean theorem.

If opposite = 5 and hypotenuse = 13, then adjacent = 132−52=169−25=12\sqrt{13^2 - 5^2} = \sqrt{169 - 25} = 12, so cos⁡(θ)=12/13\cos(\theta) = 12/13.

💡 Missing side? Pythagorean theorem. Then ratios follow automatically.

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5. Using the Pythagorean theorem to verify side relationships and find missing ratios

Using the Pythagorean Theorem to Verify Side Relationships and Find Missing Ratios

The Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) relates the three sides of any right triangle, enabling calculation of unknown sides when two are known. This directly supports finding trigonometric ratios when only partial information is available.

Intuition: Since trigonometric ratios depend on side lengths, the Pythagorean theorem bridges the gap between known and unknown quantities.

Core Rules:

  • Given any two sides, solve a2+b2=c2a^2 + b^2 = c^2 for the third side.
  • Once all sides are known, compute any trigonometric ratio directly from the definitions.
  • Verify consistency: sin⁡2(θ)+cos⁡2(θ)=1\sin^2(\theta) + \cos^2(\theta) = 1 follows from the Pythagorean theorem.

Consequence: This method ensures accurate ratio calculations without requiring angle measures, relying solely on geometric properties.

Example: If opposite =3= 3 and adjacent =4= 4, then hypotenuse =32+42=5= \sqrt{3^2 + 4^2} = 5, so sin⁡(θ)=35\sin(\theta) = \frac{3}{5}.

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A right triangle has an opposite side of length 6 and an adjacent side of length 8 relative to angle \theta. Calculate the value of \sin(\theta) as a decimal.

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✖️ 6. Applications: Calculating basic gradients, slopes, and resolving static force components in physics

🛠️ Real-World Applications

  • Slope/gradient of a ramp or hill equals tan⁡(θ)\tan(\theta) where θ\theta is the angle of incline.
  • To find the horizontal component of a force, use Fcos⁡(θ)F \cos(\theta).
  • To find the vertical component of a force, use Fsin⁡(θ)F \sin(\theta).
  • Engineers use these ratios to resolve forces into perpendicular directions.
  • Architects use tangent to calculate roof pitch and stair angles.

A 100 Newton force at 30 degrees to horizontal has horizontal component 100cos⁡(30∘)=100⋅32≈86.6100 \cos(30^\circ) = 100 \cdot \frac{\sqrt{3}}{2} \approx 86.6 N and vertical component 100sin⁡(30∘)=50100 \sin(30^\circ) = 50 N.

💡 Cosine gives horizontal, sine gives vertical—split any diagonal force.

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6. Applications: Calculating basic gradients, slopes, and resolving static force components in physics

Applications: Calculating Basic Gradients, Slopes, and Resolving Static Force Components in Physics

Trigonometric ratios model real-world phenomena where angles relate to proportional quantities. Slope or gradient of an incline equals tan⁡(θ)\tan(\theta) where θ\theta is the angle of elevation. In physics, forces resolve into perpendicular components using sine and cosine.

Intuition: Tangent naturally describes steepness (rise over run), while sine and cosine decompose vectors along coordinate axes.

Core Rules:

  • Slope: m=tan⁡(θ)=riserunm = \tan(\theta) = \frac{\text{rise}}{\text{run}}
  • Force components: Horizontal component =Fcos⁡(θ)= F \cos(\theta), vertical component =Fsin⁡(θ)= F \sin(\theta) for force FF at angle θ\theta from horizontal.
  • These applications assume small angles or linear approximations in practical contexts.

Consequence: Trigonometry translates angular information into measurable linear quantities essential for engineering and physics.

Example: A 100 N force at 30∘30^\circ has horizontal component 100cos⁡(30∘)=100⋅32≈86.6100 \cos(30^\circ) = 100 \cdot \frac{\sqrt{3}}{2} \approx 86.6 N.

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A road has an angle of elevation of 45 degrees. What is the slope of the road?

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