Glossary

10.8 Right Triangle Trigonometry

3 min read•june 18, 2024

The is a powerful tool for solving problems and finding distances between points. It states that in a right triangle, the square of the equals the sum of the squares of the other two sides.

Special right triangles and trigonometric functions build on this foundation. 30-60-90 and 45-45-90 triangles have specific side ratios, while , , and relate angles to side lengths in right triangles.

Pythagorean Theorem and Distance

Pythagorean theorem applications

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  • States in a right triangle, the square of the hypotenuse length equals the sum of the squares of the other two side lengths a2+b2=c2a^2 + b^2 = c^2
    • aa and bb represent the lengths of the triangle's legs (shorter sides)
    • cc represents the length of the triangle's hypotenuse (longest side)
  • Find an unknown side length by substituting known values into the theorem and solving for the unknown variable
    • Given a right triangle with legs measuring 3 and 4 units, find the hypotenuse length using 32+42=c23^2 + 4^2 = c^2, solving for c=5c = 5 units
  • Calculate the distance between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) using the derived from the theorem d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
    • Find the distance between points (2, 3) and (5, 7) by plugging into the formula d=(5−2)2+(7−3)2=32+42=5d = \sqrt{(5 - 2)^2 + (7 - 3)^2} = \sqrt{3^2 + 4^2} = 5 units

Special Right Triangles and Trigonometric Functions

Special right triangle ratios

  • 30-60-90 right triangles have angle measures of 30°, 60°, and 90° with side length ratios of 1:3:21 : \sqrt{3} : 2
    • Shortest side 30° angle, longest side (hypotenuse) opposite 90° angle, remaining side opposite 60° angle
    • In a with hypotenuse 6, the side lengths are 3, 333\sqrt{3}, and 6 units
  • 45-45-90 right triangles have angle measures of 45°, 45°, and 90° with side length ratios of 1:1:21 : 1 : \sqrt{2}
    • Two shorter sides (legs) are equal in length, longest side (hypotenuse) opposite 90° angle
    • In a with legs 5, the hypotenuse is 525\sqrt{2} units
  • Solve problems by identifying the triangle type and using corresponding side length ratios to set up and solve equations
    • Given a 30-60-90 triangle with hypotenuse 10, find the shorter leg length using the ratio 1:3:21 : \sqrt{3} : 2, solving for the shorter leg 102=5\frac{10}{2} = 5 units

Trigonometric functions in triangles

  • Express relationships between angles and side lengths of a right triangle
    • Sine (sin) = opposite / hypotenuse
    • Cosine (cos) = / hypotenuse
    • Tangent (tan) = opposite / adjacent
  • Find a missing side length using the appropriate function with known angle measure and side length, then solve for the unknown side
    • Given a right triangle with hypotenuse 13 and angle 30°, find the side adjacent to the angle using cos⁡30°=x13\cos 30° = \frac{x}{13}, solving for x=13cos⁡30°≈11.3x = 13 \cos 30° \approx 11.3 units
  • Inverse functions (sin−1^{-1}, cos−1^{-1}, tan−1^{-1}) find missing angle measures given the ratio of two side lengths
    • If tan⁡θ=724\tan \theta = \frac{7}{24}, find θ\theta using θ=tan⁡−1(724)≈16.3°\theta = \tan^{-1} (\frac{7}{24}) \approx 16.3°
  • These functions can be visualized on the , which represents all possible angle measures and their corresponding trigonometric values

Advanced Trigonometric Concepts

Angle measurement and trigonometric applications

  • provides an alternative way to express angles, where one radian is the angle subtended by an arc length equal to the radius of a circle
  • The relates the ratios of sides to sines of opposite angles in any triangle, not just right triangles
  • The generalizes the for any triangle, relating the square of one side to the squares of the other two sides and the cosine of the included angle
  • are equations involving trigonometric functions that are true for all values of the variables
  • Angles of elevation and depression are used in practical applications to measure the angle between the horizontal and the line of sight to an object above or below the observer, respectively

Key Terms to Review (23)

30-60-90 triangle: A 30-60-90 triangle is a specific type of right triangle where the angles measure 30 degrees, 60 degrees, and 90 degrees. This triangle has a unique ratio of side lengths, which makes it useful for various applications in trigonometry. The relationships between the angles and sides in this triangle allow for easy calculations involving sine, cosine, and tangent, linking it closely to right triangle trigonometry.
45-45-90 triangle: A 45-45-90 triangle is a specific type of right triangle where the two non-hypotenuse angles are each 45 degrees. This unique configuration leads to a distinct relationship between the lengths of its sides, which is essential in understanding right triangle trigonometry. The side lengths of a 45-45-90 triangle follow the ratio of $1 : 1 : \sqrt{2}$, making it easier to solve for missing side lengths and apply in various mathematical contexts.
Adjacent: In geometry, 'adjacent' refers to two sides or angles that are next to each other. In right triangle trigonometry, this term is crucial as it helps identify the sides related to a given angle and the relationships between them. Understanding which side is adjacent allows for accurate calculations involving sine, cosine, and tangent functions, which are fundamental in solving problems involving right triangles.
Angle of depression: The angle of depression is the angle formed between a horizontal line and the line of sight from an observer to an object located below the horizontal line. This concept is crucial in right triangle trigonometry as it helps in determining distances and heights by using trigonometric ratios. The angle of depression is equal to the angle of elevation from the object being observed back up to the observer's horizontal line, making it a useful tool for solving various real-world problems involving heights and distances.
Angle of elevation: The angle of elevation is the angle formed by a horizontal line and the line of sight to an object above that horizontal line. This concept is crucial in right triangle trigonometry as it helps to determine heights and distances when observing objects from a lower point.
Cosine: Cosine is a trigonometric function defined for an angle in a right triangle as the ratio of the length of the adjacent side to the length of the hypotenuse. This function is essential for understanding relationships between angles and sides in right triangles, and it also extends to the unit circle, where it represents the x-coordinate of a point on the circle corresponding to a given angle.
Distance Formula: The distance formula is a mathematical equation used to determine the distance between two points in a coordinate plane. This formula connects algebra with geometry, particularly in right triangle trigonometry, where it derives from the Pythagorean theorem. By calculating the distance, it helps in solving problems involving triangles and understanding relationships between angles and sides.
Hypotenuse: The hypotenuse is the longest side of a right triangle, opposite the right angle. It plays a crucial role in various mathematical concepts, especially in trigonometry and geometry, where it serves as a reference for calculating the lengths of the other sides using the Pythagorean theorem and trigonometric ratios.
Inverse cosine: The inverse cosine function, denoted as \( \cos^{-1}(x) \) or \( \arccos(x) \), is used to find the angle whose cosine is a given value. This function is essential in right triangle trigonometry as it allows for the calculation of angles when the lengths of the sides of a triangle are known. Understanding inverse cosine is crucial for solving triangles, especially when determining unknown angles using the ratios of the sides.
Inverse sine: Inverse sine, often denoted as 'sin^{-1}(x)' or 'arcsin(x)', is a mathematical function that determines the angle whose sine value is 'x'. In the context of right triangle trigonometry, inverse sine allows us to find the angles of a triangle when given the lengths of the opposite side and the hypotenuse. It is crucial for solving problems involving right triangles, as it connects the sides of a triangle to its angles, making it essential for calculations in various applications like physics and engineering.
Inverse tangent: The inverse tangent, often written as \( \tan^{-1}(x) \) or \( \text{arctan}(x) \), is a trigonometric function that determines the angle whose tangent is a given number. It connects the ratio of the opposite side to the adjacent side in a right triangle to the angle itself, allowing you to find angles when you know the lengths of the sides. This function is essential for solving various problems involving right triangles and is widely used in trigonometry to determine angle measures.
Law of Cosines: The Law of Cosines is a fundamental formula used in trigonometry that relates the lengths of the sides of a triangle to the cosine of one of its angles. This formula is especially useful for solving triangles that are not right triangles, allowing the calculation of unknown side lengths or angle measures based on known values. It serves as a crucial tool for connecting side lengths and angles in various types of triangles, expanding the application of trigonometric principles beyond just right triangles.
Law of Sines: The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. This mathematical relationship is crucial for solving triangles, especially when dealing with non-right triangles, as it allows us to find unknown lengths and angles using known values.
Opposite: In right triangle trigonometry, the term 'opposite' refers to the side of a right triangle that is directly across from a given angle. This is an essential concept in understanding relationships between the angles and sides of the triangle, particularly when using trigonometric ratios such as sine and tangent. The opposite side is crucial for calculating the heights or lengths in various practical applications involving right triangles.
Pythagorean Theorem: The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be written as \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse and \(a\) and \(b\) are the other two sides.
Pythagorean theorem: The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This relationship connects geometry and algebra, forming the basis for understanding distances and relationships within triangular structures and applications in real-world scenarios.
Radian measure: Radian measure is a way to quantify angles based on the radius of a circle, where one radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius of that circle. This form of measuring angles provides a direct relationship between the angle and the arc length, making it particularly useful in trigonometry, especially in understanding right triangles and their properties. Radian measure is crucial for various applications in mathematics, physics, and engineering, as it simplifies many formulas involving angles.
Right triangle: A right triangle is a type of triangle that has one angle measuring exactly 90 degrees, which is known as the right angle. This unique feature allows right triangles to follow specific properties and relationships that are essential in geometry and trigonometry, particularly involving the lengths of their sides and the angles formed between them. The presence of the right angle makes right triangles fundamental for various applications, including calculations related to distance, height, and angles.
Sine: Sine is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the length of the hypotenuse. It is used extensively in various mathematical contexts, especially when analyzing periodic phenomena, waveforms, and in navigation. Sine values can be expressed in both rational and irrational forms, particularly when dealing with angles that do not correspond to simple fractions.
SOHCAHTOA: SOHCAHTOA is a mnemonic device used to remember the definitions of the sine, cosine, and tangent functions in relation to right triangles. It stands for Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, and Tangent = Opposite/Adjacent. This simple tool is essential for solving various problems involving right triangles and trigonometric functions.
Tangent: A tangent is a line that touches a curve at a single point without crossing it. In mathematics, this concept is crucial in understanding angles and slopes, especially in the context of circles and right triangles. Tangents also play a role in various fields such as calculus and trigonometry, particularly when dealing with the relationships between angles and side lengths in triangles.
Trigonometric Identities: Trigonometric identities are equations that relate the angles and ratios of a right triangle through trigonometric functions like sine, cosine, and tangent. These identities are fundamental tools in solving various problems involving triangles, providing essential relationships between different trigonometric functions that simplify calculations and help in proving other mathematical statements.
Unit Circle: The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is a fundamental concept in trigonometry that provides a geometric representation of the sine, cosine, and tangent functions, making it an essential tool for understanding right triangle relationships and angles in standard position.
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