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Chapter 9: Problem 16

Given \(\triangle \mathrm{ABC}\) with \(\angle \mathrm{C}=90^{\circ}\), indicate whether each statement is true Always (A), Sometimes (S), or Never (N). a. \(\sin \angle A=\cos \angle B\) b. \(\sin \angle A=\tan \angle A\) c. \(\sin \angle A=\cos \angle A\)

Short Answer

Expert verified
a. Always. b. Sometimes. c. Sometimes.

Step by step solution

01

Analyzing Statement a

Statement a says that \( \sin \angle A=\cos \angle B \). Recall that in a right triangle, the sine of an angle is the opposite side length over the hypotenuse, and the cosine of an angle is the adjacent side length over the hypotenuse. But, \(\angle A\) and \(\angle B\) are complementary angles in a right triangle, which means that the side opposite to \(\angle A\) is adjacent to \(\angle B\) and vice versa. So, \( \sin \angle A \) will always equal to \( \cos \angle B \). Therefore, statement a is always true.
02

Analyzing Statement b

Statement b says that \( \sin \angle A=\tan \angle A \). Recall that the tangent of an angle in a right triangle is the opposite side length over the adjacent side length. The sine of an angle, however, is the opposite side length over the hypotenuse. For these two values to be equal, the hypotenuse would have to be the same length as the adjacent side, which is only possible if \(\angle A = 45^{\circ}\). Therefore, statement b is sometimes true.
03

Analyzing Statement c

Statement c says that \( \sin \angle A=\cos \angle A \). Recall that the sine of an angle in a right triangle is the opposite side length over the hypotenuse, while the cosine is the adjacent side length over the hypotenuse. These two will be equal only if the opposite side equals the adjacent side, which is only possible if \(\angle A = 45^{\circ}\). Therefore, statement c is also sometimes true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complementary Angles
In a right triangle, the non-right angles are complementary, which means they add up to 90 degrees. If \( riangle \text{ABC}\) has a right angle at \( ext{C}\), then \( ext{A} + \text{B} = 90^\circ\). This relationship gives rise to interesting connections between trigonometric functions.
For instance, the side opposite to \( ext{A}\) is adjacent to \( ext{B}\), and similarly, the side adjacent to \( ext{A}\) is opposite to \( ext{B}\). Because of this, the sine of one angle will equal the cosine of its complement.
  • For \( ext{sin} \angle \text{A}\), you measure the length of the side opposite to \( ext{A}\) over the hypotenuse.
  • For \( ext{cos} \angle \text{B}\), you measure the adjacent side (or opposite to \( ext{A}\)) over the hypotenuse.
Thus, \( ext{sin} \angle \text{A} = \cos \angle \text{B}\) holds true always in such triangles.
Trigonometric Ratios
Trigonometric ratios help us relate the angles of a triangle to its side lengths. These ratios are fundamentally important in right triangles. The three primary trigonometric ratios are sine, cosine, and tangent.
The sine of an angle \( heta\) in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse. It helps us understand how the height (opposite side) scales with the hypotenuse.
  • \(\text{sin} \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
The tangent of an angle is the ratio of the length of the opposite side to the adjacent side. This is useful in gauging how the rise compares to the run across a triangle.
  • \(\text{tan} \theta = \frac{\text{Opposite}}{\text{Adjacent}}\)
These relationships form the basis for solving various geometrical problems involving triangles.
Sine and Cosine Equality
The sine and cosine of an angle in a right triangle will only be equal when the triangle is isosceles, specifically when the angle is \({45^\circ}\). In this special triangle, the opposite and adjacent sides are of equal length.
Let's break it down with the term for sine and cosine:
  • \(\text{sin} \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
  • \(\text{cos} \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
If the opposite side and adjacent side are equal, as they would be in a \({45^\circ}\) angle, these ratios become equal. Thus, \( ext{sin} \angle A = \cos \angle A\) sometimes holds true.
This condition emphasizes the unique symmetry of an isosceles right triangle, where both non-right angles measure \({45^\circ}\).

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Most popular questions from this chapter

Woody Woodpecker pecked at a 17 -m wooden pole until it cracked and the upper part fell, with the top hitting the ground \(10 \mathrm{m}\) from the foot of the pole. since the upper part had not completely broken off, Woody pecked away where the pole had cracked. How far was Woody above the ground?

ABCDEFGH is a rectangular solid. a If face diagonal CH measures \(17,\) edge GH measures \(8,\) and edge FG measures \(6,\) how long is diagonal \(\overline{\mathrm{AG}} ?\) b If diagonal \(\overline{\mathrm{AG}}\) measures \(50,\) edge \(\overline{\mathrm{AE}}\) measures \(40,\) and edge EF measures 3 how long is edge FG? (GRAPH CANT COPY)

Given: Regular square pyramid ABCDE, with slant height \(\overline{\mathrm{AF}},\) altitude \(\overline{\mathrm{AG}}\), and base BCDE: perimeter of BCDE \(=40\) $$ \angle \mathrm{AFG}=60^{\circ} $$ Find: The altitude and the slant height (GRAPH CANT COPY)

Rhombus PQRS has a perimeter of 60 and one diagonal of \(15 .\) Find the two possible values of \(\sin \angle \mathrm{PQS}\)

Use a tree diagram to solve the following problem:. A bag contains eight right triangles. Five are members of the \((3,\) 4\. 5 ) family, and two are \(30^{\circ}-60^{\circ}-90^{\circ}\) triangles. A puppy falls over the bag, and two triangles fall out on the floor. a What is the probability that both are members of the \((3,4,5)\) family? What is the probability that at least one of the triangles is a member of the \((3,4,5)\) family? c. What is the probability that one is a member of the \((3,4,5)\) family and the other is a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle?
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