What is the definition of sine in a right triangle?

In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. For example, if angle θ is opposite side 'a' and the hypotenuse is 'c', then sin(θ) = a/c.

What is the Pythagorean theorem and how does it relate to trigonometry?

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). It can be expressed as c² = a² + b². This theorem is foundational in trigonometry, as it relates the sides of a triangle to the sine, cosine, and tangent functions.

How do you calculate the cosine of an angle in a right triangle?

The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. For angle θ, if the adjacent side is 'b' and the hypotenuse is 'c', then cos(θ) = b/c.

If tan(θ) = 3/4, what are the values of sin(θ) and cos(θ)?

Hint: Use the definition of tangent and the Pythagorean identity.

Since tan(θ) = opposite/adjacent = 3/4, we can think of a right triangle where the opposite side is 3 and the adjacent side is 4. Using the Pythagorean theorem: hypotenuse (c) = √(3² + 4²) = √(9 + 16) = √25 = 5. Thus, sin(θ) = opposite/hypotenuse = 3/5 and cos(θ) = adjacent/hypotenuse = 4/5.

What is the relationship between the angles in a right triangle and their trigonometric ratios?

The trigonometric ratios (sine, cosine, and tangent) are defined based on the angles in a right triangle. For a given angle θ, the sine, cosine, and tangent can be expressed as: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, and tan(θ) = opposite/adjacent. These ratios remain consistent regardless of the triangle's size, and they are used to find unknown sides or angles in various problems.

What are the six trigonometric functions?

The six trigonometric functions are: sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec), and cotangent (cot). They are defined as follows: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent, cosec(θ) = hypotenuse/opposite, sec(θ) = hypotenuse/adjacent, cot(θ) = adjacent/opposite.

What is the value of sin(30°)?

The value of sin(30°) is 1/2.

Given a triangle where AB = 5, BC = 12, find sin C where C is the angle opposite side AB.

Hint: Use the Pythagorean theorem to find the hypotenuse.

To find sin C, we need the length of the hypotenuse (AC). Using the Pythagorean theorem: AC = √(AB² + BC²) = √(5² + 12²) = √(25 + 144) = √169 = 13. Therefore, sin C = opposite/hypotenuse = 5/13.

What is the cosine of 45°?

The value of cos(45°) is 1/√2.

If sec(θ) = 2, what is cos(θ)?

Hint: Recall that sec(θ) is the reciprocal of cos(θ).

Since sec(θ) = 1/cos(θ), if sec(θ) = 2, then cos(θ) = 1/2.

Find the angle θ if sin(θ) = 1/√2.

Hint: Consider common angles in trigonometry.

The angle θ such that sin(θ) = 1/√2 is 45°.

Using the sine rule, if a = 10, b = 15, and angle A = 30°, find angle B.

Hint: Apply the sine rule: a/sin(A) = b/sin(B).

Using the sine rule, we have 10/sin(30°) = 15/sin(B). Since sin(30°) = 1/2, we find 10/(1/2) = 15/sin(B) => 20 = 15/sin(B) => sin(B) = 15/20 = 3/4. Therefore, angle B = sin⁻¹(3/4).

What is the relationship between the angles in a triangle using the sine law?

The sine law states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. It can be expressed as a/sin(A) = b/sin(B) = c/sin(C).

If cot(θ) = 1, what are the values of sin(θ) and cos(θ)?

Hint: Recall that cot(θ) is cos(θ)/sin(θ).

If cot(θ) = 1, then sin(θ) = cos(θ). The only angle where this holds true in the range [0°, 90°] is θ = 45°. Therefore, sin(45°) = cos(45°) = 1/√2.

What is the definition of sine in a right triangle?

In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. For example, if angle θ is opposite side 'a' and the hypotenuse is 'c', then sin(θ) = a/c.