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Questions and Answers
Cosine is defined as the ratio of the length of the adjacent side to the length of the opposite side.
Cosine is defined as the ratio of the length of the adjacent side to the length of the opposite side.
False
Tangent is defined as the ratio of the length of the opposite side to the length of the adjacent side.
Tangent is defined as the ratio of the length of the opposite side to the length of the adjacent side.
True
Cotangent is the reciprocal of sine.
Cotangent is the reciprocal of sine.
False
The Pythagorean identity is used to derive the trigonometric ratios.
The Pythagorean identity is used to derive the trigonometric ratios.
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Secant is the ratio of the length of the opposite side to the length of the hypotenuse.
Secant is the ratio of the length of the opposite side to the length of the hypotenuse.
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The Pythagorean identity is only true for acute angles.
The Pythagorean identity is only true for acute angles.
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The tangent of an angle is equal to the sine of the angle divided by the cosine of the angle.
The tangent of an angle is equal to the sine of the angle divided by the cosine of the angle.
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The Pythagorean identity can be used to convert between sine and tangent values of an angle.
The Pythagorean identity can be used to convert between sine and tangent values of an angle.
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The cotangent of an angle is equal to the reciprocal of the tangent of the angle.
The cotangent of an angle is equal to the reciprocal of the tangent of the angle.
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The Pythagorean identity states that the sum of the sine and cosine of an angle is equal to 1.
The Pythagorean identity states that the sum of the sine and cosine of an angle is equal to 1.
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Study Notes
Trigonometric Ratios and the Pythagorean Identity
Trigonometric ratios are fundamental concepts in mathematics that are used to relate the sides of a right triangle to the angles of the triangle. The three most commonly used trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). In addition to these, there are three more ratios: cotangent (cot), secant (sec), and cosecant (csc). These ratios can be derived from the Pythagorean theorem and the Pythagorean identity.
Sine, Cosine, and Tangent
Sine (sin) is defined as the ratio of the length of the opposite side to the length of the hypotenuse. It represents the ratio of the height of a right triangle to its base.
Cosine (cos) is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. It represents the ratio of the base of a right triangle to its height.
Tangent (tan) is defined as the ratio of the length of the opposite side to the length of the adjacent side. It represents the ratio of the height of a right triangle to its base.
Cotangent, Secant, and Cosecant
Cotangent (cot) is the reciprocal of tangent. It is the ratio of the length of the adjacent side to the length of the opposite side.
Secant (sec) is the reciprocal of cosine. It is the ratio of the length of the hypotenuse to the length of the adjacent side.
Cosecant (csc) is the reciprocal of sine. It is the ratio of the length of the hypotenuse to the length of the opposite side.
Pythagorean Identity
The Pythagorean identity is a fundamental theorem in mathematics that states that the sum of the square of the sine of an angle and the square of the cosine of an angle is equal to 1. It can be written as:
sin²θ + cos²θ = 1
This identity is true for all real values of θ. It is a result of applying the Pythagorean theorem on the right triangle that is formed in the unit circle for each angle θ. The Pythagorean theorem states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the other two sides. In the case of the unit circle, the hypotenuse, or the radius, is 1, and the adjacent and opposite sides are sin(θ) and cos(θ), respectively. Therefore, the Pythagorean theorem becomes:
1² = (sin²(θ)) + (cos²(θ))
Squaring both sides, we get:
sin²(θ) + cos²(θ) = 1
This is the Pythagorean identity.
Applications of Pythagorean Identity
The Pythagorean identity is a powerful tool in mathematics that can be used to solve various problems involving trigonometric functions. It can be used to convert between sine and cosine values of an angle without knowing the angle itself. For example, if we know the sine and cosine values of an angle, we can use the Pythagorean identity to find the other value.
The Pythagorean identity can also be used to prove other trigonometric identities, such as the reciprocal identities and quotient identities.
Reciprocal Identities
Reciprocal identities are equations that relate the sine, cosine, tangent, cotangent, secant, and cosecant functions to their reciprocals. They are as follows:
sin(θ) = 1/csc(θ) cos(θ) = 1/sec(θ) tan(θ) = 1/cot(θ) cot(θ) = 1/tan(θ) sec(θ) = 1/cos(θ) csc(θ) = 1/sin(θ)
Quotient Identities
Quotient identities are equations that relate the tangent and cotangent functions to the sine and cosine functions. They are as follows:
tan(θ) = sin(θ)/cos(θ) cot(θ) = cos(θ)/sin(θ)
Proofs using Pythagorean Identity
The Pythagorean identity can be used to prove various trigonometric identities. For example, it can be used to prove the quotient identities:
- tan(θ) = sin(θ)/cos(θ)
- cot(θ) = cos(θ)/sin(θ)
To prove these identities, we can use the Pythagorean identity:
sin²(θ) + cos²(θ) = 1
Dividing both sides by sin(θ)cos(θ), we get:
(sin(θ)/cos(θ))² + (cos(θ)/sin(θ))² = 1/sin(θ)cos(θ)
Simplifying, we get:
tan²(θ) + cot²(θ) = 1
Comparing this with the Pythagorean identity, we can see that:
tan(θ) = sin(θ)/cos(θ) cot(θ) = cos(θ)/sin(θ)
These are the quotient identities.
Double Angle and Half Angle Formulas
The Pythagorean identity can also be used to derive the double angle and half angle formulas for sine and cosine. The double angle formulas are:
sin(2θ) = 2sin(θ)cos(θ) cos(2θ) = cos²(θ) - sin²(θ)
The half angle formulas are:
sin(θ/2) = ±√((1 - cos(θ))/2) cos(θ/2) = ±√((1 + cos(θ))/2)
These formulas are useful in various applications, such as solving trigonometric equations and finding the values of trigonometric functions for given angles.
In conclusion, the Pythagorean identity is a fundamental theorem in mathematics that relates the sine, cosine, and tangent functions to their reciprocals and quotients. It is a powerful tool that can be used to solve various problems involving trigon
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Description
Test your understanding of trigonometric ratios, including sine, cosine, and tangent, and how they relate to the Pythagorean identity. Learn about the applications of the Pythagorean identity in solving trigonometric problems and proving identities.
