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Calculating Cosine of an Angle

The cosine (cos) of an angle in a right-angled triangle is a trigonometric function that represents the ratio of the length of the adjacent side to the hypotenuse. The cosine function is generally used to find unknown side lengths or angles in right triangles.

The cosine of an angle \( \theta \) is defined as:

\( \cos(\theta) = \dfrac{\text{length of adjacent side}}{\text{length of hypotenuse}} \)

where:

• adjacent side is the side adjacent to angle \( \theta \), excluding the hypotenuse.
• hypotenuse is the side opposite to the right angle.

Consider the following triangle ABC, right angled at vertex B.

For the angle \( \theta \) at vertex A:

• Adjacent side = AB
• Hypotenuse = AC

Therefore,

\( \cos(\theta) = \dfrac{\text{length of adjacent side}}{\text{length of hypotenuse}} \)\( = \dfrac{\text{length of side AB}}{\text{length of side AC}} \)

Examples

The following examples cover how to use the cosine of an angle to find the length of the adjacent side or the hypotenuse in different practical scenarios.

1. What is the cosine of an angle θ if the length of the adjacent side is 4 units and the length of the hypotenuse is 5 units?

Answer

Given:

• Length of adjacent side = 4 units
• Length of hypotenuse = 5 units

The formula to calculate the cosine of an angle θ is:

\( \cos(\theta) = \dfrac{\text{length of adjacent side}}{\text{length of hypotenuse}} \)

Substitute the given values into the formula:

\( \cos(\theta) = \dfrac{4}{5} \)

Simplify the expression:

\( \cos(\theta) = 0.8 \)

∴ cos(θ) = 0.8

2. A person is walking up a hill that forms a 30° angle with the horizontal. If the person covers a distance of 100 meters along the hill (hypotenuse), how much horizontal distance have they covered?

Answer

Given:

• Distance along the hill (hypotenuse) = 100 meters
• Angle with the horizontal θ = 30°

Using the cosine function to find the horizontal distance (adjacent side):

\( \cos(30°) = \dfrac{\text{horizontal distance}}{100} \)

Simplify the expression:

\( 0.866 = \dfrac{\text{horizontal distance}}{100} \)

Now solve for the horizontal distance:

\( \text{Horizontal distance} = 0.866 \times 100 = 86.6 \text{ meters} \)

∴ The person has covered 86.6 meters horizontally.

3. A ladder is leaning against a wall, making a 45° angle with the ground. If the length of the ladder is 10 meters, how far is the base of the ladder from the wall?

Answer

Given:

• Length of the ladder (hypotenuse) = 10 meters
• Angle with the ground θ = 45°

Using the cosine function to find the distance of the base of the ladder from the wall (adjacent side):

\( \cos(45°) = \dfrac{\text{distance from the wall}}{10} \)

Simplify the expression:

\( 0.707 = \dfrac{\text{distance from the wall}}{10} \)

Now solve for the distance from the wall:

\( \text{Distance from the wall} = 0.707 \times 10 = 7.07 \text{ meters} \)

∴ The base of the ladder is 7.07 meters from the wall.

{
  "topic": "cos",
  "function": "cos",
  "function_desc": "Cosine",
  "x_symbol": "θ",
  "x_desc": "angle",
  "category": "Trigonometry",
  "formula": "cos(θ) = (Length of Adjacent side) / (Length of Hypotenuse)",
  "formula_in_js": "Math.cos(θ)",
  "description": "In a right angled triangle, Cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.",
  "template": "trigonometry_function",
  "content": "<h2>Calculating Cosine of an Angle</h2>\n<p>The cosine (cos) of an angle in a right-angled triangle is a trigonometric function that represents the ratio of the length of the adjacent side to the hypotenuse. The cosine function is generally used to find unknown side lengths or angles in right triangles.</p>\n<p>The cosine of an angle \\( \\theta \\) is defined as:</p>\n<p>\\( \\cos(\\theta) = \\dfrac{\\text{length of adjacent side}}{\\text{length of hypotenuse}} \\)</p>\n<p>where:</p>\n<ul>\n<li><b>adjacent side</b> is the side adjacent to angle \\( \\theta \\), excluding the hypotenuse.</li>\n<li><b>hypotenuse</b> is the side opposite to the right angle.</li>\n</ul><p>Consider the following triangle ABC, right angled at vertex B.</p><img src=\"/images/mathematics/trigonometry.webp\" alt=\"Right Angled Triangle to Calculate cos of angle\" width=\"256\" height=\"256\" loading=\"lazy\"><p>For the angle \\( \\theta \\) at vertex A:</p><ul><li>Adjacent side = AB</li><li>Hypotenuse = AC</li></ul><p>Therefore,</p><p>\\( \\cos(\\theta) = \\dfrac{\\text{length of adjacent side}}{\\text{length of hypotenuse}} \\)\\( = \\dfrac{\\text{length of side AB}}{\\text{length of side AC}} \\)</p>\n\n<h2>Examples</h2><p>The following examples cover how to use the cosine of an angle to find the length of the adjacent side or the hypotenuse in different practical scenarios.</p><div class=\"example\"><h3 class=\"question\"><span class=\"example_n\">1.</span> What is the cosine of an angle θ if the length of the adjacent side is 4 units and the length of the hypotenuse is 5 units?</h3><h4 class=\"answer\">Answer</h4><p>Given:</p><ul><li>Length of adjacent side = 4 units</li><li>Length of hypotenuse = 5 units</li></ul><p>The formula to calculate the cosine of an angle θ is:</p><p class=\"tabspace\">\\( \\cos(\\theta) = \\dfrac{\\text{length of adjacent side}}{\\text{length of hypotenuse}} \\)</p><p>Substitute the given values into the formula:</p><p class=\"tabspace\">\\( \\cos(\\theta) = \\dfrac{4}{5} \\)</p><p>Simplify the expression:</p><p class=\"tabspace\">\\( \\cos(\\theta) = 0.8 \\)</p><p class=\"tabspace answer\">∴ cos(θ) = 0.8</p></div><div class=\"example\"><h3 class=\"question\"><span class=\"example_n\">2.</span> A person is walking up a hill that forms a 30° angle with the horizontal. If the person covers a distance of 100 meters along the hill (hypotenuse), how much horizontal distance have they covered?</h3><h4 class=\"answer\">Answer</h4><p>Given:</p><ul><li>Distance along the hill (hypotenuse) = 100 meters</li><li>Angle with the horizontal θ = 30°</li></ul><p>Using the cosine function to find the horizontal distance (adjacent side):</p><p class=\"tabspace\">\\( \\cos(30°) = \\dfrac{\\text{horizontal distance}}{100} \\)</p><p>Simplify the expression:</p><p class=\"tabspace\">\\( 0.866 = \\dfrac{\\text{horizontal distance}}{100} \\)</p><p>Now solve for the horizontal distance:</p><p class=\"tabspace\">\\( \\text{Horizontal distance} = 0.866 \\times 100 = 86.6 \\text{ meters} \\)</p><p class=\"tabspace answer\">∴ The person has covered 86.6 meters horizontally.</p></div><div class=\"example\"><h3 class=\"question\"><span class=\"example_n\">3.</span> A ladder is leaning against a wall, making a 45° angle with the ground. If the length of the ladder is 10 meters, how far is the base of the ladder from the wall?</h3><h4 class=\"answer\">Answer</h4><p>Given:</p><ul><li>Length of the ladder (hypotenuse) = 10 meters</li><li>Angle with the ground θ = 45°</li></ul><p>Using the cosine function to find the distance of the base of the ladder from the wall (adjacent side):</p><p class=\"tabspace\">\\( \\cos(45°) = \\dfrac{\\text{distance from the wall}}{10} \\)</p><p>Simplify the expression:</p><p class=\"tabspace\">\\( 0.707 = \\dfrac{\\text{distance from the wall}}{10} \\)</p><p>Now solve for the distance from the wall:</p><p class=\"tabspace\">\\( \\text{Distance from the wall} = 0.707 \\times 10 = 7.07 \\text{ meters} \\)</p><p class=\"tabspace answer\">∴ The base of the ladder is 7.07 meters from the wall.</p></div>"
}