Retake: Unit 3 Laws, Area, and Trig Identities

Front

What is the formula for the area of a triangle?

Back

The area of a triangle can be calculated using the formula: \( A = \frac{1}{2} \times base \times height \).

Front

What is the relationship between sine, cosine, and tangent in a right triangle?

Back

In a right triangle, the relationships are: \( \sin(x) = \frac{opposite}{hypotenuse} \), \( \cos(x) = \frac{adjacent}{hypotenuse} \), and \( \tan(x) = \frac{opposite}{adjacent} = \frac{\sin(x)}{\cos(x)} \).

Front

What is the Pythagorean theorem?

Back

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): \( c^2 = a^2 + b^2 \).

Front

How do you simplify \( tan(B) \cdot (cot(B) + tan(B)) \)?

Back

Using the identity \( cot(B) = \frac{1}{tan(B)} \), the expression simplifies to \( tan(B) \cdot (\frac{1}{tan(B)} + tan(B)) = 1 + tan^2(B) = sec^2(B) \).

Front

What is the exact value of \( tan(22.5^{\circ}) \)?

Back

The exact value of \( tan(22.5^{\circ}) \) is \( \sqrt{2} - 1 \).

Front

What is the Law of Sines?

Back

The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant: \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \).

Front

What is the Law of Cosines?

Back

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles: \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \).

Front

How do you find the area of a triangle given two sides and the included angle?

Back

The area can be calculated using the formula: \( A = \frac{1}{2} \times a \times b \times \sin(C) \), where a and b are the lengths of the sides and C is the included angle.

Front

What is the definition of cotangent?

Back

Cotangent is defined as the reciprocal of tangent: \( cot(x) = \frac{1}{tan(x)} = \frac{adjacent}{opposite} \).

Front

What is the relationship between secant and cosine?

Back

Secant is the reciprocal of cosine: \( sec(x) = \frac{1}{cos(x)} \).