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Tangent is found by dividing the measure of the ______ side by the measure of the adjacent side.
opposite
Secant is calculated using the formula sec(θ) = hypotenuse / ______ side.
opposite
Cotangent is calculated using the formula cot(θ) = adjacent side / ______ side.
opposite
The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two shorter sides is equal to the square of the ______ side.
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Trigonometry provides methods for solving problems involving right triangles, where one angle is known and all three sides must be calculated using trigonometric ratios, such as sine, cosine, and ______.
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Trigonometry is the branch of mathematics dealing with relations between sides and angles of ______
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The six basic trigonometric functions include sine (sin), cosine (cos), tangent (tan), secant, cosecant (csc), and ______
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Sine is calculated by dividing the measure of the opposite side by the measure of the ______
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Cosine is calculated using the formula cos(θ) = adjacent side / ______
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Trigonometry is crucial for navigators, engineers, architects, and other professionals who need to determine distances, heights, or angles related to their ______
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Study Notes
Trigonometry
Trigonometry is the branch of mathematics dealing with relations between sides and angles of triangles. It provides methods for calculating distances, heights, and other measurements based on known information in these three-sided shapes. This field is crucial for navigators, engineers, architects, and other professionals who need to determine distances, heights, or angles related to their work. In this article, we will explore trigonometry's essential concepts, including trigonometric ratios, the Pythagorean theorem, solving right triangles, applied problems, and angles of elevation and depression.
Trigonometric Ratios
The six basic trigonometric functions—sine (sin), cosine (cos), tangent (tan), secant, cosecant (csc), and cotangent (cot)—are used to describe the relationships between sides and angles in a triangle. These functions relate the lengths of the sides opposite, adjacent, or hypotenuse to the angle being measured. For example, the sine function gives the ratio of the length of one side to the longest side forming a right angle.
Sine (sin)
Sine is calculated by dividing the measure of the opposite side by the measure of the hypotenuse, so sin(θ) = opposite side / hypotenuse. Here, c represents the length of the side opposite the angle, and b is the length of the side adjacent to the angle.
Cosine (cos)
Cosine is calculated using the formula cos(θ) = adjacent side / hypotenuse. In this case, a represents the length of the side adjacent to the angle, and b is the length of the side opposite the angle.
Tangent (tan)
Tangent is found by dividing the measure of the opposite side by the measure of the adjacent side, so tan(θ) = opposite side / adjacent side. Here, c represents the length of the side opposite the angle, and a is the length of the side adjacent to the angle.
Secant (sec)
Secant is calculated using the formula sec(θ) = hypotenuse / opposite side. In this case, c represents the length of the side opposite the angle, and b is the length of the hypotenuse.
Cosecant (csc)
Cosecant is found by dividing the measure of the hypotenuse by the measure of the opposite side, so csc(θ) = hypotenuse / opposite side. Here, c represents the length of the side opposite the angle, and b is the length of the hypotenuse.
Cotangent (cot)
Cotangent is calculated using the formula cot(θ) = adjacent side / opposite side. In this case, a represents the length of the side adjacent to the angle, and c is the length of the side opposite the angle.
Pythagorean Theorem
The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two shorter sides is equal to the square of the longest side. Mathematically, it can be written as: a² + b² = c². Here, c represents the length of the hypotenuse, while a and b represent the lengths of the legs adjacent to the right angle.
Solving Right Triangles
Trigonometry provides methods for solving problems involving right triangles, where one angle is known and all three sides must be calculated. Trigonometric ratios, such as sine, cosine, and tangent, along with the Pythagorean theorem, can be used to determine the unknown side lengths or angles.
Applied Problems
Trigonometry has numerous applications in various fields, such as architecture, engineering, and navigation. For example, it can be used to calculate the height of a tall building or the distance across a river, as well as to determine the angles at which a rooftop solar panel should be installed.
Angles of Elevation and Depression
Trigonometric ratios can be used to calculate angles of elevation and depression, which are the angles formed between a horizontal line and a line extending from the eye to an object. For example, if you know the height of an object and the angle of elevation or depression, you can use trigonometry to determine the distance to the object.
In conclusion, trigonometry is a fundamental branch of mathematics that provides methods for calculating distances, heights, and angles related to triangles. Its applications are vast and diverse, making it an essential tool for many professionals. Through the use of trigonometric ratios, the Pythagorean theorem, and the ability to solve right triangles, trigonometry can help us understand the world around us and navigate through various challenges.
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Description
Test your knowledge of trigonometry essentials, including trigonometric ratios, the Pythagorean theorem, solving right triangles, applied problems, and angles of elevation and depression. Explore key concepts that are crucial for understanding and applying trigonometry in various fields.
