What is the definition of sine in a right triangle?

In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. For example, if angle A is the angle of interest, then sin(A) = opposite/hypotenuse.

What is the cosine of an angle in a right triangle?

The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. For example, cos(A) = adjacent/hypotenuse.

How is the tangent of an angle defined in terms of sine and cosine?

The tangent of an angle is defined as the ratio of the sine of that angle to the cosine of that angle. Mathematically, this is expressed as tan(A) = sin(A)/cos(A).

If sin(θ) = 0.6, what is cos(θ) using the Pythagorean identity? Hint: Use the identity sin²(θ) + cos²(θ) = 1.

Using the Pythagorean identity: sin²(θ) + cos²(θ) = 1. First, calculate sin²(θ) = 0.6² = 0.36. Then, cos²(θ) = 1 - 0.36 = 0.64. Taking the square root gives cos(θ) = ±√0.64 = ±0.8.

Solve for x if tan(x) = 1. Hint: Think about the angles where the tangent is equal to 1.

The tangent of an angle is equal to 1 at x = 45° (or π/4 radians) and also at x = 225° (or 5π/4 radians) in the interval [0°, 360°].

What is the acronym to remember the definitions of sine, cosine, and tangent?

The acronym is SOHCAHTOA, which stands for: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

In a right triangle with an angle A, if the opposite side is 3 and the hypotenuse is 5, what is sin(A)?

Sin(A) = opposite/hypotenuse = 3/5.

If cos(A) = 0.8, what is sin(A) using the Pythagorean identity? Hint: Use sin²(A) + cos²(A) = 1.

Using the Pythagorean identity: sin²(A) + cos²(A) = 1. Calculate cos²(A) = 0.8² = 0.64. Then, sin²(A) = 1 - 0.64 = 0.36. Taking the square root gives sin(A) = ±√0.36 = ±0.6.

What is tan(30°)? Hint: Remember the special angles and their ratios.

Tan(30°) = 1/√3 or √3/3.

What is the relationship between sin(A) and sin(90° - A)?

Sin(A) = cos(90° - A). This shows that sine and cosine are complementary.

If in triangle ABC, angle B is a right angle, and side AC (hypotenuse) is 10 and side AB (adjacent to angle A) is 6, what is cos(A)?

Cos(A) = adjacent/hypotenuse = 6/10 = 3/5.

What is the value of sin(90°)?

Sin(90°) = 1.

If tan(B) = 3/4, what are the lengths of the opposite and adjacent sides if the hypotenuse is 5? Hint: Use the tangent ratio.

Using tan(B) = opposite/adjacent, let opposite = 3k and adjacent = 4k. Then, (3k)² + (4k)² = 5². This gives 9k² + 16k² = 25, so 25k² = 25, hence k = 1. Thus, opposite = 3 and adjacent = 4.

What is the value of cos(0°)?

Cos(0°) = 1.

In a right triangle, if the lengths of the sides are 5, 12, and 13, what is sin(A) if A is the angle opposite the side of length 12? Hint: Identify the opposite and hypotenuse.

Sin(A) = opposite/hypotenuse = 12/13.

If sin(θ) = ½, what are the possible values of θ in the interval [0°, 360°]? Hint: Consider the unit circle.

The possible values of θ are 30° and 150°.

Determine the value of tan(45°).

Tan(45°) = 1.

If a triangle has sides of lengths 8, 15, and 17, what is cos(C) where C is the angle opposite the side of length 15? Hint: Use the cosine ratio.

Cos(C) = adjacent/hypotenuse = 8/17.

What is the sine of a 30° angle?

Sin(30°) = ½.

If angle A is 60° in a right triangle, what is the value of tan(60°)?

Tan(60°) = √3.