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4.3.2 Sine, Cosine, and Tangent Ratios

Defining and calculating the three primary trigonometric ratios (sin, cos, tan) using opposite, adjacent, and hypotenuse sides.

定义

The sine, cosine, and tangent ratios are the three primary trigonometric ratios used to relate the angles and sides of a right triangle. For a right triangle with an acute angle \(\theta\), these ratios are defined as follows:

Sine (sin): The ratio of the length of the side opposite to angle \(\theta\) to the length of the hypotenuse. Denoted as \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\).

Cosine (cos): The ratio of the length of the side adjacent to angle \(\theta\) to the length of the hypotenuse. Denoted as \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\).

Tangent (tan): The ratio of the length of the side opposite to angle \(\theta\) to the length of the side adjacent to angle \(\theta\). Denoted as \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\). \nA common mnemonic device to remember these definitions is SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. \nThese ratios depend only on the measure of the angle \(\theta\), not on the size of the triangle, making them fundamental tools for solving problems involving right triangles and angles.

核心公式

  • \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\)
  • \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\)
  • \(\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\)
  • \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)
  • \(\sin^2(\theta) + \cos^2(\theta) = 1\)

易错点

  • ⚠️ ["Confusing which side is opposite and which is adjacent: Students often forget that the opposite and adjacent sides depend on which angle is being considered. The hypotenuse is always opposite the right angle, but opposite and adjacent are relative to the angle in question.", "Incorrectly identifying the hypotenuse: Some students mistakenly use a leg of the triangle as the hypotenuse. The hypotenuse is always the longest side and is always opposite the right angle (90°).", "Mixing up the ratios: Students may reverse the numerator and denominator, such as writing \(\sin(\theta) = \frac{\text{hypotenuse}}{\text{opposite}}\) instead of the correct \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\).", "Forgetting to use the correct angle: When a triangle has multiple angles, students sometimes calculate the ratio for the wrong angle, especially when the angle is not clearly labeled or when working with complementary angles."]