Sin, Cos, Tan Practice Quiz

The sine function represents the ratio of the opposite side to the hypotenuse. Since sin(90°) equals 1, this is a fundamental trigonometric value.

Using the sine function, sin(30°) equals 0.5. Multiplying the hypotenuse (10) by 0.5 gives 5, which is the length of the side opposite the 30° angle.

The Pythagorean identity expresses the relationship between sine and cosine as sin²θ + cos²θ = 1. This identity is fundamental to trigonometry.

In a right triangle with equal legs, the tangent of 45° is the ratio of the opposite side to the adjacent side, which equals 1. This is a standard value in trigonometry.

The sine function of 0° is 0 because the length of the side opposite a 0° angle is zero. This value forms a basic building block in trigonometric functions.

Using the identity cosθ = √(1 - sin²θ), substituting sinθ = 0.6 gives cosθ = √(1 - 0.36) = √0.64 = 0.8. This demonstrates the interdependence of sine and cosine.

Cosine of 60° is 0.5 in the first quadrant. Recognizing standard trigonometric values helps in quickly solving this problem.

Tangent is defined as the ratio of the length of the opposite side to the adjacent side in a right triangle. This definition is crucial for solving many trigonometric problems.

The tangent of an angle is defined as opposite/adjacent. Multiplying 1.5 by 4 gives the length of the opposite side as 6.

The sine of 30° is a well-known trigonometric value, equal to 0.5. This value is derived from the ratios in a 30-60-90 triangle.

Using the sine function, sin(60°) equals √3/2. Multiplying the hypotenuse (12) by √3/2 yields 6√3, which is the correct length.

With tanθ = 3/4, a right triangle can be formed with sides 3, 4, and 5. Thus, sinθ, which is opposite/hypotenuse, equals 3/5.

Using the identity cosθ = √(1 - sin²θ), substituting sinθ = 0.8 gives cosθ = √(1 - 0.64) = √0.36 = 0.6. This reflects the complementary nature of sine and cosine.

The reciprocal of the sine function is cosecant, denoted as cscθ, and is defined as 1/sinθ. This is one of the fundamental reciprocal identities in trigonometry.

Since tanθ is defined as sinθ/cosθ, and sin(0°) is 0 while cos(0°) is 1, tan(0°) equals 0. This is a basic and important trigonometric fact.

First, use the identity cosθ = √(1 - sin²θ) to find cosθ = √(1 - (4/9)) = √(5/9) = √5/3. Then, tanθ = sinθ/cosθ equals (2/3) divided by (√5/3), which simplifies to 2/√5 and rationalizes to 2√5/5.

With tanθ defined as opposite/adjacent, the opposite side is 3 à - 4 = 12. Applying the Pythagorean theorem gives the hypotenuse as √(12² + 4²) = √(144 + 16) = √160 = 4√10.

Using the identity cosθ = √(1 - sin²θ), plug in sinθ = 0.7 to get cosθ = √(1 - 0.49) = √0.51, which approximates to 0.71 when rounded to the nearest hundredth.

The double-angle formula for sine is given by sin2θ = 2sinθcosθ. This formula is widely used in simplifying trigonometric expressions and solving equations.

Cosθ is defined as adjacent/hypotenuse, so the hypotenuse is 8/0.8 = 10. Then, using sinθ = √(1 - cos²θ) = √(1 - 0.64) = 0.6, the opposite side is 0.6 à - 10 = 6.