Equivalent fractions

Learning Objectives Prove that trigonometric ratios for a given angle are constant by demonstrating the equivalent fractions generated by similar right-angled triangles. Convert between degree and radian angle measures by setting up and solving proportions based on equivalent fractions. Calculate arc lengths and sector areas by applying the principle of equivalent fractions relating parts of a circle to the whole. Connect the slope of a line (rise/run) to the tangent of its angle of inclination as an application of equivalent fractional ratios. Manipulate trigonometric expressions into equivalent fractional forms to verify trigonometric identities. Analyze and solve problems involving angular velocity by using equivalent ratios of angle to time. Why does sin(30°) always equa...

TermDefinitionExample Trigonometric RatioA ratio of the lengths of two sides in a right-angled triangle, corresponding to a specific acute angle (θ). These ratios are fractions that remain constant for a given angle, regardless of the triangle's size.For an angle θ, sin(θ) = Opposite / Hypotenuse. If Opposite = 3 and Hypotenuse = 5, the ratio is 3/5. In a larger similar triangle, the sides might be 6 and 10, but the ratio 6/10 is an equivalent fraction to 3/5. Similar TrianglesTriangles that have the same shape because their corresponding angles are equal. The ratio of their corresponding side lengths is constant, forming a set of equivalent fractions.If ΔABC ~ ΔXYZ, then AB/XY = BC/YZ = AC/XZ. This property is why sin(θ) is the same in all right triangles containing angle θ. RadianA...

Trigonometric Ratios from Similar Triangles \frac{\text{opposite}_1}{\text{hypotenuse}_1} = \frac{\text{opposite}_2}{\text{hypotenuse}_2} \implies \sin(\theta)_1 = \sin(\theta)_2 For any two right triangles with a shared acute angle θ, they are similar. This means the ratios of their corresponding sides (the trigonometric ratios) form equivalent fractions. This is why the value of sin(θ), cos(θ), and tan(θ) depends only on the angle, not the size of the triangle. Degree-Radian Conversion Proportion \frac{D}{180^\circ} = \frac{R}{\pi} This formula sets up an equality of two ratios (equivalent fractions) to convert between an angle measured in degrees (D) and the same angle measured in radians (R). It relates a part of the semi-circle to the whole semi-circle in both units....