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Chapter 11: Problem 13

Use a calculator to find the indicated cosine ratio to four decimal places. $$\cos 90^{\circ}$$

Short Answer

Expert verified
\( \cos 90^{\circ} = 0 \).

Step by step solution

01

Understand the cosine function

The cosine function, denoted as \( \cos \theta \), measures the ratio of the adjacent side to the hypotenuse in a right-angled triangle for an angle \( \theta \). It is also defined on the unit circle as the x-coordinate of the point that corresponds to the angle \( \theta \).
02

Identify the specified angle

In this problem, the angle given is \( 90^{\circ} \). On the unit circle, this angle is located at the positive y-axis.
03

Find the cosine of 90 degrees

On the unit circle, the coordinates for \( 90^{\circ} \) are \((0, 1)\). The cosine is the x-coordinate of this point. Therefore, \( \cos 90^{\circ} = 0 \).
04

Confirm with a calculator

To ensure accuracy, use a calculator. Enter \( \cos 90 \) and verify that the calculator reads out 0. This helps confirm our finding using the unit circle.

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Key Concepts

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

Trigonometric Ratios
Trigonometric ratios are fundamental in understanding relationships in right-angled triangles and the unit circle. The primary trigonometric ratios include sine, cosine, and tangent, each defined by the sides of a right-angled triangle:
  • Sine (\( \sin \theta \)): The ratio of the opposite side to the hypotenuse.
  • Cosine (\( \cos \theta \)): The ratio of the adjacent side to the hypotenuse.
  • Tangent (\( \tan \theta \)): The ratio of the opposite side to the adjacent side.
These ratios are chiefly useful in solving triangles, calculating angles, and encapsulating the functionalities of periodic phenomena such as waves and oscillations. Understanding these ratios is essential for studying geometry, trigonometry, and many applications in physics and engineering. Whenever you use these ratios, remember that they provide a bridge between geometric shapes and algebraic functions.
Unit Circle
The unit circle is a powerful mathematical tool that marries geometry and trigonometry. It's a circle with a radius of one, centered at the origin of a coordinate system. This simple structure offers a wealth of insights into trigonometric functions.
The circle allows for visualization of angles and their corresponding sine and cosine values:
  • Each angle \( \theta \) on the unit circle corresponds to a point \((x, y)\).
  • The \(x\) coordinate represents the cosine of the angle (\( \cos \theta \)).
  • The \(y\) coordinate represents the sine of the angle (\( \sin \theta \)).
At \(90^{\circ}\), the process becomes particularly easy. The point lies at (0, 1), making \( \cos 90^{\circ} = 0 \) and \( \sin 90^{\circ} = 1 \). This graphical approach simplifies understanding and aids quickly finding cosine and sine values of standard angles without complicated calculations.
Right-Angled Triangles
Right-angled triangles are the cornerstone of trigonometry. With one angle fixed at \(90^{\circ}\), these triangles allow us to apply trigonometric ratios effectively.
The essential properties include:
  • The side opposite the right-angle is the hypotenuse鈥攖he longest side.
  • The two other sides are the 'adjacent' and 'opposite' sides, depending on the angle in question.
  • Right-angled triangles are the foundation for defining trigonometric functions such as sine, cosine, and tangent.
Angles in these triangles, when measured in degrees or radians, reveal the intrinsic properties of the sides. For example, if you know two sides, you can find any other angle or side using trigonometric ratios. Understanding right-angled triangles' structure simplifies complex geometry and is essential for solving real-world engineering and physics problems.

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Most popular questions from this chapter

Angle measures should be given to the nearest degree; distances should be given to the nearest tenth of a unit. In searching for survivors of a boating accident, a helicopter moves horizontally across the ocean at an altitude of \(200 \mathrm{ft}\) above the water. If a man clinging to a life raft is seen through an angle of depression of \(12^{\circ},\) what is the distance from the helicopter to the man in the water?

In Exercises 7 to \(14,\) use either Table 11.2 or a calculator to find the sine of the indicated angle to four decimal places. $$\sin 0^{\circ}$$

In regular pentagon \(A B C D E,\) sides \(\overline{A B}\) and \(\overline{B C}\) along with diagonal \(\overline{A C}\) form isosceles \(\triangle A B C .\) Let \(A B=B C=s\) In terms of \(s,\) find an expression for a) \(h,\) the length of the altitude of \(\triangle A B C\) from vertex \(B\) to side \(\overrightarrow{A C}\). b) \(d,\) the length of diagonal \(\overline{A C}\) of regular pentagon ABCDE.

In Exercises 7 to \(14,\) use either Table 11.2 or a calculator to find the sine of the indicated angle to four decimal places. $$\sin 23^{\circ}$$

Use the Law of Sines or the Law of cosines to solve each problem. Angle measures should be found to the nearest degree and areas and distances to the nearest tenth of a unit. A triangular lot has street dimensions of \(150 \mathrm{ft}\) and \(180 \mathrm{ft}\) and an included angle of \(80^{\circ}\) for these two sides. a) Find the length of the remaining side of the lot. b) Find the area of the lot in square feet.
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