8.4 Right-Triangle Trigonometry, the Unit Circle, and Radian Measure

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1 Chapter 1: ACCUPLACER Suite Overview & Testing Mechanics

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2 Chapter 2: Next-Generation Reading

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6 Chapter 6: Next-Generation Quantitative Reasoning, Algebra, and Statistics (QAS)

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7 Chapter 7: Next-Generation Advanced Algebra, Quadratics & Polynomials

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8 Chapter 8: Functions, Exponentials, Logarithms & Trigonometry

8.1 Function Notation, Transformations, Compositions & Inverses 8.2 Domain, Range, and Interpreting Functions in Context 8.3 Exponential & Logarithmic Functions, Laws & Modeling 8.4 Right-Triangle Trigonometry, the Unit Circle, and Radian Measure 8.5 Law of Sines, Law of Cosines, and Trigonometric Equations 8.6 Choosing Your Math Test: Arithmetic, QAS, or AAF

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8.4 Right-Triangle Trigonometry, the Unit Circle, and Radian Measure

Key Takeaways Explain This Topic AI

Right triangle trigonometric ratios define sine, cosine, and tangent as ratios of side lengths via SOH CAH TOA (sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj), complemented by reciprocal ratios cosecant (csc θ = 1/sin θ), secant (sec θ = 1/cos θ), and cotangent (cot θ = 1/tan θ).

Special right triangles provide exact trigonometric values without a calculator: 45°-45°-90° triangles exhibit side ratios 1 : 1 : √2, and 30°-60°-90° triangles exhibit side ratios 1 : √3 : 2.

Radian measure relates angle size to arc length on a unit circle (360° = 2π rad); conversions require multiplying degrees by π/180°, enabling arc length calculation s = r·θ and sector area calculation A = (1/2)r²·θ (where θ is strictly in radians).

On the unit circle (x² + y² = 1), any angle θ corresponds to coordinates (cos θ, sin θ); the ASTC rule (All Students Take Calculus) identifies positive function signs across quadrants, paired with reference angles θ' in [0, π/2].

The fundamental Pythagorean identity sin²θ + cos²θ = 1 and its variants (1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ) allow algebraic simplification of trigonometric expressions; sinusoidal graphs y = A·sin(B(x - C)) + D possess amplitude |A|, period T = 2π/B, horizontal phase shift C, and vertical midline y = D.

Last updated: August 2026

Trigonometric Ratios, Unit Circle, Radian Measure & Circles

Core Advanced Mathematics: Trigonometry on the ACCUPLACER Advanced Algebra and Functions (AAF) assessment links geometric right triangle side ratios, circular radian dynamics, unit circle coordinates across all four quadrants, fundamental Pythagorean identities, and periodic sinusoidal wave models. Mastery requires computing exact trigonometric ratios, executing angle and sector conversions, applying identities, and analyzing wave parameters.

Right Triangle Trigonometry (SOH CAH TOA) & Reciprocal Functions

In a right triangle with an acute angle θ \theta θ , side lengths are classified relative to θ \theta θ as the Opposite leg ( O O O ) , the Adjacent leg ( A A A ) , and the Hypotenuse ( H H H ) (the side opposite the 90 ∘ 90^\circ 9 0 ∘ right angle). Right Triangle Trigonometry Reference

• ▲
• ╱│
• ╱ │
• Hypotenuse (H) ╱ │ Opposite (O)
• ╱ │
• ╱ │
• ╱ θ │
• ───────┴─
• Adjacent (A)

Primary Trigonometric Ratios (SOH CAH TOA) sin ⁡ θ = Opposite Hypotenuse = O H cos ⁡ θ = Adjacent Hypotenuse = A H tan ⁡ θ = Opposite Adjacent = O A \sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{O}{H} \qquad \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{A}{H} \qquad \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{O}{A} sin θ = Hypotenuse Opposite ​ = H O ​ cos θ = Hypotenuse Adjacent ​ = H A ​ tan θ = Adjacent Opposite ​ = A O ​

Reciprocal Trigonometric Ratios

Each primary ratio has a corresponding reciprocal trigonometric function: csc ⁡ θ = Hypotenuse Opposite = 1 sin ⁡ θ sec ⁡ θ = Hypotenuse Adjacent = 1 cos ⁡ θ cot ⁡ θ = Adjacent Opposite = 1 tan ⁡ θ \csc\theta = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{1}{\tan\theta} csc θ = Opposite Hypotenuse ​ = sin θ 1 ​ sec θ = Adjacent Hypotenuse ​ = cos θ 1 ​ cot θ = Opposite Adjacent ​ = tan θ 1 ​

Co-Function Identities (Complementary Angles)

Because the two acute angles in a right triangle sum to 90 ∘ 90^\circ 9 0 ∘ (or π 2 \frac{\pi}{2} 2 π ​ radians), co-functions of complementary angles are equal:

sin ⁡ ( 90 ∘ − θ ) = cos ⁡ θ \sin(90^\circ - \theta) = \cos\theta sin ( 9 0 ∘ − θ ) = cos θ and cos ⁡ ( 90 ∘ − θ ) = sin ⁡ θ \cos(90^\circ - \theta) = \sin\theta cos ( 9 0 ∘ − θ ) = sin θ

tan ⁡ ( 90 ∘ − θ ) = cot ⁡ θ \tan(90^\circ - \theta) = \cot\theta tan ( 9 0 ∘ − θ ) = cot θ and cot ⁡ ( 90 ∘ − θ ) = tan ⁡ θ \cot(90^\circ - \theta) = \tan\theta cot ( 9 0 ∘ − θ ) = tan θ

sec ⁡ ( 90 ∘ − θ ) = csc ⁡ θ \sec(90^\circ - \theta) = \csc\theta sec ( 9 0 ∘ − θ ) = csc θ and csc ⁡ ( 90 ∘ − θ ) = sec ⁡ θ \csc(90^\circ - \theta) = \sec\theta csc ( 9 0 ∘ − θ ) = sec θ

Special Right Triangles & Exact Trigonometric Values

Exact trigonometric values for benchmark angles ( 30 ∘ , 45 ∘ , 60 ∘ 30^\circ, 45^\circ, 60^\circ 3 0 ∘ , 4 5 ∘ , 6 0 ∘ ) derive directly from two geometric special right triangles.

1. The 45 ∘ − 45 ∘ − 90 ∘ 45^\circ-45^\circ-90^\circ 4 5 ∘ − 4 5 ∘ − 9 0 ∘ ( π 4 − π 4 − π 2 \frac{\pi}{4}-\frac{\pi}{4}-\frac{\pi}{2} 4 π ​ − 4 π ​ − 2 π ​ ) Isosceles Right Triangle

• Side Ratio: 1 : 1 : 2 1 : 1 : \sqrt{2} 1 : 1 : 2 ​ (Leg : Leg : Hypotenuse)
• sin ⁡ 45 ∘ = 1 2 = 2 2 \sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} sin 4 5 ∘ = 2 ​ 1 ​ = 2 2 ​ ​
• cos ⁡ 45 ∘ = 1 2 = 2 2 \cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} cos 4 5 ∘ = 2 ​ 1 ​ = 2 2 ​ ​
• tan ⁡ 45 ∘ = 1 1 = 1 \tan 45^\circ = \frac{1}{1} = 1 tan 4 5 ∘ = 1 1 ​ = 1

2. The 30 ∘ − 60 ∘ − 90 ∘ 30^\circ-60^\circ-90^\circ 3 0 ∘ − 6 0 ∘ − 9 0 ∘ ( π 6 − π 3 − π 2 \frac{\pi}{6}-\frac{\pi}{3}-\frac{\pi}{2} 6 π ​ − 3 π ​ − 2 π ​ ) Scalene Right Triangle

Side Ratio: 1 : 3 : 2 1 : \sqrt{3} : 2 1 : 3 ​ : 2 (Short Leg opposite 30 ∘ 30^\circ 3 0 ∘ : Long Leg opposite 60 ∘ 60^\circ 6 0 ∘ : Hypotenuse)

sin ⁡ 30 ∘ = 1 2 cos ⁡ 30 ∘ = 3 2 tan ⁡ 30 ∘ = 1 3 = 3 3 \sin 30^\circ = \frac{1}{2} \qquad \cos 30^\circ = \frac{\sqrt{3}}{2} \qquad \tan 30^\circ = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} sin 3 0 ∘ = 2 1 ​ cos 3 0 ∘ = 2 3 ​ ​ tan 3 0 ∘ = 3 ​ 1 ​ = 3 3 ​ ​

sin ⁡ 60 ∘ = 3 2 cos ⁡ 60 ∘ = 1 2 tan ⁡ 60 ∘ = 3 1 = 3 \sin 60^\circ = \frac{\sqrt{3}}{2} \qquad \cos 60^\circ = \frac{1}{2} \qquad \tan 60^\circ = \frac{\sqrt{3}}{1} = \sqrt{3} sin 6 0 ∘ = 2 3 ​ ​ cos 6 0 ∘ = 2 1 ​ tan 6 0 ∘ = 1 3 ​ ​ = 3 ​

Radian Measure, Arc Length & Sector Area

A radian is the measure of a central angle θ \theta θ that subtends a circular arc equal in length to the radius r r r of the circle. 1 full revolution = 360 ∘ = 2 π radians ⟹ 180 ∘ = π radians 1\text{ full revolution} = 360^\circ = 2\pi\text{ radians} \implies 180^\circ = \pi\text{ radians} 1 full revolution = 36 0 ∘ = 2 π radians ⟹ 18 0 ∘ = π radians

Angle Conversion Formulas

Degrees to Radians: Multiply by π 180 ∘ \frac{\pi}{180^\circ} 18 0 ∘ π ​ (e.g., 135 ∘ × π 180 ∘ = 3 π 4 rad 135^\circ \times \frac{\pi}{180^\circ} = \frac{3\pi}{4}\text{ rad} 13 5 ∘ × 18 0 ∘ π ​ = 4 3 π ​ rad )

Radians to Degrees: Multiply by 180 ∘ π \frac{180^\circ}{\pi} π 18 0 ∘ ​ (e.g., 2 π 3 rad × 180 ∘ π = 120 ∘ \frac{2\pi}{3}\text{ rad} \times \frac{180^\circ}{\pi} = 120^\circ 3 2 π ​ rad × π 18 0 ∘ ​ = 12 0 ∘ )

Circle Geometry Formulas with Radians

For a circle of radius r r r with central angle θ \theta θ strictly measured in radians : +-----------------------------------------------------------------------------------------+

• | CIRCULAR SECTOR & ARC FORMULAS |
• +-----------------------+---------------------------------+-------------------------------+
• | Geometric Measure | Formula (θ in Radians) | Formula (θ in Degrees) |
• +-----------------------+---------------------------------+-------------------------------+
• | Arc Length (s) | s = r · θ | s = (θ / 360°) · 2πr |
• | Sector Area (A) | A = (1/2) · r² · θ | A = (θ / 360°) · πr² |
• +-----------------------+---------------------------------+-------------------------------+
• Worked Example: Arc Length and Sector Area

A circular sector in a circle of radius r = 6 cm r = 6\text{ cm} r = 6 cm is subtended by a central angle of 120 ∘ 120^\circ 12 0 ∘ . Find the arc length s s s and sector area A A A .

Step 1 (Convert to radians): θ = 120 ∘ × π 180 ∘ = 2 π 3 radians \theta = 120^\circ \times \frac{\pi}{180^\circ} = \frac{2\pi}{3}\text{ radians} θ = 12 0 ∘ × 18 0 ∘ π ​ = 3 2 π ​ radians .

Step 2 (Calculate arc length): s = r ⋅ θ = 6 × 2 π 3 = 4 π cm ≈ 12.57 cm s = r \cdot \theta = 6 \times \frac{2\pi}{3} = 4\pi\text{ cm} \approx 12.57\text{ cm} s = r ⋅ θ = 6 × 3 2 π ​ = 4 π cm ≈ 12.57 cm .

Step 3 (Calculate sector area): A = 1 2 r 2 θ = 1 2 ( 6 ) 2 ( 2 π 3 ) = 1 2 ( 36 ) ( 2 π 3 ) = 18 × 2 π 3 = 12 π cm 2 ≈ 37.70 cm 2 A = \frac{1}{2} r^2 \theta = \frac{1}{2} (6)^2 \left(\frac{2\pi}{3}\right) = \frac{1}{2} (36) \left(\frac{2\pi}{3}\right) = 18 \times \frac{2\pi}{3} = 12\pi\text{ cm}^2 \approx 37.70\text{ cm}^2 A = 2 1 ​ r 2 θ = 2 1 ​ ( 6 ) 2 ( 3 2 π ​ ) = 2 1 ​ ( 36 ) ( 3 2 π ​ ) = 18 × 3 2 π ​ = 12 π cm 2 ≈ 37.70 cm 2 .

The Unit Circle & Quadrant Trigonometry

The Unit Circle is a circle centered at the origin ( 0 , 0 ) (0, 0) ( 0 , 0 ) with a radius of exactly r = 1 r = 1 r = 1 , governed by the algebraic equation: x 2 + y 2 = 1 x^2 + y^2 = 1 x 2 + y 2 = 1

For any angle θ \theta θ drawn in standard position (vertex at origin, initial ray along positive x x x -axis), the terminal ray intersects the unit circle at coordinates: ( x , y ) = ( cos ⁡ θ , sin ⁡ θ ) and tan ⁡ θ = y x = sin ⁡ θ cos ⁡ θ ( x ≠ 0 ) (x, y) = (\cos\theta, \sin\theta) \quad \text{and} \quad \tan\theta = \frac{y}{x} = \frac{\sin\theta}{\cos\theta} \; (x \neq 0) ( x , y ) = ( cos θ , sin θ ) and tan θ = x y ​ = cos θ sin θ ​ ( x  = 0 ) The Unit Circle Quadrants

• y ▲ (0, 1) [90°, π/2]
• │
• QUADRANT II │ QUADRANT I
• (-x, +y) │ (+x, +y)
• Sine & Csc (+) │ ALL Functions (+)
• [180°, π] │ [0°, 0 rad]
• (-1, 0) ────────────────────────┼────────────────────────► (1, 0) x
• │ [360°, 2π]
• QUADRANT III │ QUADRANT IV
• (-x, -y) │ (+x, -y)
• Tan & Cot (+) │ Cosine & Sec (+)
• │
• ▼ (0, -1) [270°, 3π/2]
• Quadrant Signs Rule (ASTC: "All Students Take Calculus")

Quadrant I ( 0 < θ < π 2 0 < \theta < \frac{\pi}{2} 0 < θ < 2 π ​ ): A ll trigonometric functions are positive .

Quadrant II ( π 2 < θ < π \frac{\pi}{2} < \theta < \pi 2 π ​ < θ < π ): S ine (and cosecant) are positive ; cosine and tangent are negative.

Quadrant III ( π < θ < 3 π 2 \pi < \theta < \frac{3\pi}{2} π < θ < 2 3 π ​ ): T angent (and cotangent) are positive ; sine and cosine are negative.

Quadrant IV ( 3 π 2 < θ < 2 π \frac{3\pi}{2} < \theta < 2\pi 2 3 π ​ < θ < 2 π ): C osine (and secant) are positive ; sine and tangent are negative.

Reference Angles ( θ ′ \theta' θ ′ )

The reference angle θ ′ \theta' θ ′ is the positive acute angle formed between the terminal side of θ \theta θ and the x x x -axis:

• Quadrant of θ \theta θ Reference Angle Formula (Degrees) Reference Angle Formula (Radians)
• Quadrant I θ ′ = θ \theta' = \theta θ ′ = θ θ ′ = θ \theta' = \theta θ ′ = θ

Quadrant II θ ′ = 180 ∘ − θ \theta' = 180^\circ - \theta θ ′ = 18 0 ∘ − θ θ ′ = π − θ \theta' = \pi - \theta θ ′ = π − θ

Quadrant III θ ′ = θ − 180 ∘ \theta' = \theta - 180^\circ θ ′ = θ − 18 0 ∘ θ ′ = θ − π \theta' = \theta - \pi θ ′ = θ − π

Quadrant IV θ ′ = 360 ∘ − θ \theta' = 360^\circ - \theta θ ′ = 36 0 ∘ − θ θ ′ = 2 π − θ \theta' = 2\pi - \theta θ ′ = 2 π − θ

Master Unit Circle Exact Values Reference Table

Angle (Degrees) Angle (Radians) Quadrant Point Coordinates ( x , y ) = ( cos ⁡ θ , sin ⁡ θ ) (x, y) = (\cos\theta, \sin\theta) ( x , y ) = ( cos θ , sin θ ) tan ⁡ θ = sin ⁡ θ cos ⁡ θ \tan\theta = \frac{\sin\theta}{\cos\theta} tan θ = c o s θ s i n θ ​

0 ∘ 0^\circ 0 ∘ 0 0 0 Positive x x x -axis ( 1 , 0 ) (1, 0) ( 1 , 0 ) 0 0 0

30 ∘ 30^\circ 3 0 ∘ π 6 \frac{\pi}{6} 6 π ​ Quadrant I ( 3 2 , 1 2 ) \left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right) ( 2 3 ​ ​ , 2 1 ​ ) 3 3 \frac{\sqrt{3}}{3} 3 3 ​ ​

45 ∘ 45^\circ 4 5 ∘ π 4 \frac{\pi}{4} 4 π ​ Quadrant I ( 2 2 , 2 2 ) \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) ( 2 2 ​ ​ , 2 2 ​ ​ ) 1 1 1

60 ∘ 60^\circ 6 0 ∘ π 3 \frac{\pi}{3} 3 π ​ Quadrant I ( 1 2 , 3 2 ) \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right) ( 2 1 ​ , 2 3 ​ ​ ) 3 \sqrt{3} 3 ​

90 ∘ 90^\circ 9 0 ∘ π 2 \frac{\pi}{2} 2 π ​ Positive y y y -axis ( 0 , 1 ) (0, 1) ( 0 , 1 ) Undefined

120 ∘ 120^\circ 12 0 ∘ 2 π 3 \frac{2\pi}{3} 3 2 π ​ Quadrant II ( − 1 2 , 3 2 ) \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right) ( − 2 1 ​ , 2 3 ​ ​ ) − 3 -\sqrt{3} − 3 ​

135 ∘ 135^\circ 13 5 ∘ 3 π 4 \frac{3\pi}{4} 4 3 π ​ Quadrant II ( − 2 2 , 2 2 ) \left(-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) ( − 2 2 ​ ​ , 2 2 ​ ​ ) − 1 -1 − 1

150 ∘ 150^\circ 15 0 ∘ 5 π 6 \frac{5\pi}{6} 6 5 π ​ Quadrant II ( − 3 2 , 1 2 ) \left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right) ( − 2 3 ​ ​ , 2 1 ​ ) − 3 3 -\frac{\sqrt{3}}{3} − 3 3 ​ ​

180 ∘ 180^\circ 18 0 ∘ π \pi π Negative x x x -axis ( − 1 , 0 ) (-1, 0) ( − 1 , 0 ) 0 0 0

210 ∘ 210^\circ 21 0 ∘ 7 π 6 \frac{7\pi}{6} 6 7 π ​ Quadrant III ( − 3 2 , − 1 2 ) \left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right) ( − 2 3 ​ ​ , − 2 1 ​ ) 3 3 \frac{\sqrt{3}}{3} 3 3 ​ ​

225 ∘ 225^\circ 22 5 ∘ 5 π 4 \frac{5\pi}{4} 4 5 π ​ Quadrant III ( − 2 2 , − 2 2 ) \left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right) ( − 2 2 ​ ​ , − 2 2 ​ ​ ) 1 1 1

240 ∘ 240^\circ 24 0 ∘ 4 π 3 \frac{4\pi}{3} 3 4 π ​ Quadrant III ( − 1 2 , − 3 2 ) \left(-\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) ( − 2 1 ​ , − 2 3 ​ ​ ) 3 \sqrt{3} 3 ​

270 ∘ 270^\circ 27 0 ∘ 3 π 2 \frac{3\pi}{2} 2 3 π ​ Negative y y y -axis ( 0 , − 1 ) (0, -1) ( 0 , − 1 ) Undefined

300 ∘ 300^\circ 30 0 ∘ 5 π 3 \frac{5\pi}{3} 3 5 π ​ Quadrant IV ( 1 2 , − 3 2 ) \left(\frac{1}{2}, -\frac{\sqrt{3}}{2}\right) ( 2 1 ​ , − 2 3 ​ ​ ) − 3 -\sqrt{3} − 3 ​

315 ∘ 315^\circ 31 5 ∘ 7 π 4 \frac{7\pi}{4} 4 7 π ​ Quadrant IV ( 2 2 , − 2 2 ) \left(\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right) ( 2 2 ​ ​ , − 2 2 ​ ​ ) − 1 -1 − 1

330 ∘ 330^\circ 33 0 ∘ 11 π 6 \frac{11\pi}{6} 6 11 π ​ Quadrant IV ( 3 2 , − 1 2 ) \left(\frac{\sqrt{3}}{2}, -\frac{1}{2}\right) ( 2 3 ​ ​ , − 2 1 ​ ) − 3 3 -\frac{\sqrt{3}}{3} − 3 3 ​ ​

Fundamental Trigonometric Identities

Quotient & Reciprocal Identities tan ⁡ θ = sin ⁡ θ cos ⁡ θ cot ⁡ θ = cos ⁡ θ sin ⁡ θ sec ⁡ θ = 1 cos ⁡ θ csc ⁡ θ = 1 sin ⁡ θ \tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \cot\theta = \frac{\cos\theta}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \csc\theta = \frac{1}{\sin\theta} tan θ = cos θ sin θ ​ cot θ = sin θ cos θ ​ sec θ = cos θ 1 ​ csc θ = sin θ 1 ​

The Three Pythagorean Identities

Substituting x = cos ⁡ θ x = \cos\theta x = cos θ and y = sin ⁡ θ y = \sin\theta y = sin θ into the unit circle equation x 2 + y 2 = 1 x^2 + y^2 = 1 x 2 + y 2 = 1 establishes the primary Pythagorean Identity: sin ⁡ 2 θ + cos ⁡ 2 θ = 1 \sin^2\theta + \cos^2\theta = 1 sin 2 θ + cos 2 θ = 1

Dividing this equation by cos ⁡ 2 θ \cos^2\theta cos 2 θ and sin ⁡ 2 θ \sin^2\theta sin 2 θ respectively yields two secondary Pythagorean identities: 1 + tan ⁡ 2 θ = sec ⁡ 2 θ 1 + cot ⁡ 2 θ = csc ⁡ 2 θ 1 + \tan^2\theta = \sec^2\theta \qquad 1 + \cot^2\theta = \csc^2\theta 1 + tan 2 θ = sec 2 θ 1 + cot 2 θ = csc 2 θ

Worked Example: Simplifying an Identity Expression

Simplify the trigonometric expression: 1 − cos ⁡ 2 θ sin ⁡ θ cos ⁡ θ \frac{1 - \cos^2\theta}{\sin\theta \cos\theta} s i n θ c o s θ 1 − c o s 2 θ ​ .

Step 1 (Apply Pythagorean identity): Since sin ⁡ 2 θ + cos ⁡ 2 θ = 1 \sin^2\theta + \cos^2\theta = 1 sin 2 θ + cos 2 θ = 1 , we have 1 − cos ⁡ 2 θ = sin ⁡ 2 θ 1 - \cos^2\theta = \sin^2\theta 1 − cos 2 θ = sin 2 θ .

Step 2 (Substitute and reduce): sin ⁡ 2 θ sin ⁡ θ cos ⁡ θ = sin ⁡ θ cos ⁡ θ = tan ⁡ θ \frac{\sin^2\theta}{\sin\theta \cos\theta} = \frac{\sin\theta}{\cos\theta} = \tan\theta sin θ cos θ sin 2 θ ​ = cos θ sin θ ​ = tan θ

Sinusoidal Functions & Their Graphs (Sine & Cosine)

Sinusoidal functions model periodic, oscillating phenomena (e.g., sound waves, tides, alternating current): y = A sin ⁡ ( B ( x − C ) ) + D or y = A cos ⁡ ( B ( x − C ) ) + D y = A \sin(B(x - C)) + D \quad \text{or} \quad y = A \cos(B(x - C)) + D y = A sin ( B ( x − C )) + D or y = A cos ( B ( x − C )) + D

Sinusoidal Parameters Explained

Amplitude ( ∣ A ∣ |A| ∣ A ∣ ): Half the distance between the maximum and minimum values: Amplitude = ∣ A ∣ = y max ⁡ − y min ⁡ 2 \text{Amplitude} = |A| = \frac{y_{\max} - y_{\min}}{2} Amplitude = ∣ A ∣ = 2 y m a x ​ − y m i n ​ ​ (If A < 0 A < 0 A < 0 , the graph is reflected vertically across its midline).

Period ( T T T ): The horizontal distance required to complete one full cycle: T = 2 π B ⟺ B = 2 π T T = \frac{2\pi}{B} \iff B = \frac{2\pi}{T} T = B 2 π ​ ⟺ B = T 2 π ​

Frequency ( f f f ): Number of cycles completed per unit of x x x : f = 1 T = B 2 π f = \frac{1}{T} = \frac{B}{2\pi} f = T 1 ​ = 2 π B ​

Phase Shift ( C C C ): Horizontal translation (shifts right if C > 0 C > 0 C > 0 , left if C < 0 C < 0 C < 0 ).

Vertical Shift / Midline ( y = D y = D y = D ): The horizontal centerline of oscillation: Midline y = D = y max ⁡ + y min ⁡ 2 \text{Midline } y = D = \frac{y_{\max} + y_{\min}}{2} Midline y = D = 2 y m a x ​ + y m i n ​ ​

Parent Graphs Comparison: sin ⁡ ( x ) \sin(x) sin ( x ) vs. cos ⁡ ( x ) \cos(x) cos ( x )

Feature y = sin ⁡ ( x ) y = \sin(x) y = sin ( x ) y = cos ⁡ ( x ) y = \cos(x) y = cos ( x )

y y y -Intercept ( 0 , 0 ) (0, 0) ( 0 , 0 ) (starts on midline) ( 0 , 1 ) (0, 1) ( 0 , 1 ) (starts at maximum peak)

• Period 2 π 2\pi 2 π 2 π 2\pi 2 π
• Amplitude 1 1 1 1 1 1

Domain / Range ( − ∞ , ∞ ) (-\infty, \infty) ( − ∞ , ∞ ) / [ − 1 , 1 ] [-1, 1] [ − 1 , 1 ] ( − ∞ , ∞ ) (-\infty, \infty) ( − ∞ , ∞ ) / [ − 1 , 1 ] [-1, 1] [ − 1 , 1 ]

Five Key Points ( 0 , 0 ) → ( π 2 , 1 ) → ( π , 0 ) → ( 3 π 2 , − 1 ) → ( 2 π , 0 ) (0,0) \to (\frac{\pi}{2},1) \to (\pi,0) \to (\frac{3\pi}{2},-1) \to (2\pi,0) ( 0 , 0 ) → ( 2 π ​ , 1 ) → ( π , 0 ) → ( 2 3 π ​ , − 1 ) → ( 2 π , 0 ) ( 0 , 1 ) → ( π 2 , 0 ) → ( π , − 1 ) → ( 3 π 2 , 0 ) → ( 2 π , 1 ) (0,1) \to (\frac{\pi}{2},0) \to (\pi,-1) \to (\frac{3\pi}{2},0) \to (2\pi,1) ( 0 , 1 ) → ( 2 π ​ , 0 ) → ( π , − 1 ) → ( 2 3 π ​ , 0 ) → ( 2 π , 1 )

Loading diagram... Unit Circle Quadrant Signs & Reference Angles Guide