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Questions and Answers
What is the equation involving sine and cosine that can be derived from the information provided?
What is the equation involving sine and cosine that can be derived from the information provided?
What is the formula to find the tangent of an angle θ in a right triangle?
What is the formula to find the tangent of an angle θ in a right triangle?
Which trigonometric identity involves both the tangent and secant functions?
Which trigonometric identity involves both the tangent and secant functions?
If sin(θ) = 3/5 and cos(θ) = 4/5, what is the value of tan(θ)?
If sin(θ) = 3/5 and cos(θ) = 4/5, what is the value of tan(θ)?
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What trigonometric formula can be used to solve for the remaining unknown side length of a triangle given two sides and an angle?
What trigonometric formula can be used to solve for the remaining unknown side length of a triangle given two sides and an angle?
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In a right triangle, the sine of an angle is calculated as the ratio of:
In a right triangle, the sine of an angle is calculated as the ratio of:
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Which trigonometric function in a right triangle compares the length of the adjacent side to the hypotenuse?
Which trigonometric function in a right triangle compares the length of the adjacent side to the hypotenuse?
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To find the cosine of an angle in a right triangle, you would divide the length of:
To find the cosine of an angle in a right triangle, you would divide the length of:
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What is the tangent of an angle in a right triangle defined as?
What is the tangent of an angle in a right triangle defined as?
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If the sine of an angle in a right triangle is 0.6, what would be the cosine of that angle?
If the sine of an angle in a right triangle is 0.6, what would be the cosine of that angle?
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Study Notes
Trigonometry
Trigonometry is a branch of mathematics that deals with relationships between angles and lengths of sides in a right-angled triangle. The field also includes values derived from those relationships, such as sine, cosine, and tangent. This branch of mathematics plays a crucial role in various aspects of physics, engineering, and computer science.
Sine, Cosine, Tangent
Sine
The sine function assigns to each angle θ in a right triangle the ratio of the length of the side opposite that angle to the longest side of the triangle (the hypotenuse).
Finding Sine
To find the sine of an angle θ in a right triangle, divide the length of the side opposite by the length of the hypotenuse:
sin(θ) = (opposite side)/(hypotenuse)
Cosine
The cosine function assigns to each angle θ in a right triangle the ratio of the length of the adjacent side to the longest side (the hypotenuse).
Finding Cosine
To find the cosine of an angle θ in a right triangle, divide the length of the side adjacent by the length of the hypotenuse:
cos(θ) = (adjacent side)/(hypotenuse)
Tangent
The tangent function assigns to each angle θ in a right triangle the ratio of the length of the opposite side to the length of the adjacent side. It is defined as:
tan(θ) = (opposite side)/(adjacent side)
Finding Tangent
To find the tangent of an angle θ in a right triangle, divide the length of the side opposite by the length of the side adjacent:
tan(θ) = (opposite side)/(adjacent side)
Trigonometric Identities
Trigonometry has many useful formulas called "identities," which are true for all angles θ and radii r:
- sin²θ + cos²θ = 1
- tan²θ + 1 = sec²θ
- cot²θ + 1 = cosec²θ
- sin³θ + cos³θ = 1/2
- sin(θ + θ') = sin(θ)cos(θ') + cos(θ)sin(θ')
- cos(θ + θ') = cos(θ)cos(θ') - sin(θ)sin(θ')
- tan(θ + θ') = (tan(θ) + tan(θ'))/(1 - tan(θ)tan(θ'))
Solving Triangles
Solving triangles involves applying different combinations of the three fundamental functions (sine, cosine, and tangent) to obtain equations that can be solved simultaneously to determine one or more of the unknown side or angle measures. In this process, you first identify the known quantities, then apply appropriate trigonometric ratios to get the desired unknown.
For example, if we have two acute angles and one of the sides (either one of the non-hypotenuse), we can solve the triangle using two trigonometric ratios. We usually solve for one of the angles, say θ:
sin(θ) = (opposite side) / (hypotenuse)
cos(θ) = (adjacent side) / (hypotenuse)
Given the information, we can create equations:
sin(θ) = (a) / b
cos(θ) = (b) / (c)
By multiplying these two equations together, we get:
sin(θ) * cos(θ) = 1/2
Substituting the expressions for sin(θ) and cos(θ) into the equation and simplifying, we get:
(a^2) / (bc) = 1/2
Which leads to the Law of Cosines:
a² = b² + c² - 2bc*cos(θ)
This allows us to solve for the remaining unknown side length when given the measurements of the other two sides and one angle.
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Description
Test your knowledge on trigonometry fundamentals, including the concepts of sine, cosine, and tangent, as well as trigonometric identities and solving triangles using trigonometric ratios. This quiz covers essential principles in trigonometry that are crucial in various fields like physics, engineering, and computer science.
