Right Triangles & Trigonometry Practice Quiz

A right angle measures exactly 90 degrees, which is a fundamental concept in geometry. Recognizing this measurement helps in identifying and working with right angles in various problems.

The Pythagorean Theorem establishes the relationship between the lengths of the sides in a right triangle. This theorem is essential for solving many geometric problems involving right triangles.

The primary trigonometric ratios in a right triangle are sine, cosine, and tangent. These ratios connect the angles of a triangle to the ratios of its sides, forming the basis of trigonometry.

The sine of an acute angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the hypotenuse. This definition is a cornerstone of trigonometric calculations.

A triangle with one angle measuring exactly 90° is known as a right triangle. This type of triangle has distinct properties that differentiate it from other triangles.

In any right triangle, the two acute angles must sum to 90°. Subtracting 35° from 90° gives the measure of the other acute angle, which is 55°.

By applying the Pythagorean Theorem, 3² + 4² equals 9 + 16, which is 25. The square root of 25 is 5, making 5 the length of the hypotenuse.

In a right triangle, the hypotenuse is always the longest side because it is opposite the right angle. This property is vital when applying trigonometric ratios and the Pythagorean theorem.

Using the fundamental identity sin²θ + cos²θ = 1, if cos θ is 0.8, then sin²θ = 1 - 0.64, which is 0.36. Taking the square root yields sin θ = 0.6.

In a 30-60-90 triangle, the hypotenuse is always twice as long as the shorter leg. With a shorter leg of 6, the hypotenuse must be 12.

Tangent is defined as the ratio of the opposite side to the adjacent side. A tangent value of 0.75 simplifies to a ratio of 3:4, which is the correct relationship.

The cotangent of an angle is defined as the ratio of the adjacent side to the opposite side. This is the reciprocal of the tangent function, which is central to many trigonometric problems.

In a 45°-45°-90° triangle, both legs are equal, so the cosine of 45° is the ratio of one leg to the hypotenuse, which simplifies to √2/2. This value is a standard trigonometric constant.

An isosceles right triangle has two congruent acute angles. Since the sum of the two acute angles must be 90°, each angle measures 45°.

The value of sin θ being 0.5 corresponds to an angle of 30° in standard trigonometry. This is a well-known trigonometric relationship used in solving right triangles.

Applying the Pythagorean Theorem: 7² + b² = 25², which simplifies to b² = 625 - 49 = 576. The square root of 576 is 24, making it the correct length for the other leg.

The height reached on the wall can be found by multiplying the ladder's length by the sine of 65°. Calculating 20 � - sin(65°) gives approximately 18.1 ft, which is the correct answer.

With tan θ = 7/24, assign the opposite side as 7 and the adjacent side as 24. The hypotenuse is found using the Pythagorean Theorem: √(7² + 24²) = 25. Therefore, cos θ = adjacent/hypotenuse = 24/25.

Using the half-angle formula, sin 22.5° is calculated as √((1 - cos 45°)/2). Since cos 45° is √2/2, the expression simplifies to (√(2 - √2))/2. This is the simplest radical form.

The altitude-to-hypotenuse theorem states that the square of the altitude is equal to the product of the lengths of the segments into which it divides the hypotenuse. This geometric mean relationship is essential in many proofs and applications involving right triangles.