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Chapter 5: Problem 1

Fill in the blank to complete the trigonometric identity. \(\frac{\sin u}{\cos u}=\)_____

Short Answer

Expert verified
The completed trigonometric identity is \(\frac{sin u}{cos u} = tan u\).

Step by step solution

01

Recall the Basic Trigonometric Identities

The basic trigonometric identities are sine (sin), cosine (cos), and tangent (tan). Knowing these will be necessary for this and many other trigonometry problems. The identity formula for the tangent function is \(tan u = \frac{sin u}{cos u}\).
02

Matching the identity

If you look at the given equation \(\frac{sin u}{cos u}\), you will notice it matches the right side of the tangent identity. This means that the blank should be filled with \(tan u\).

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Key Concepts

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

Sine
Sine is one of the three primary functions in trigonometry, often abbreviated as "sin". In a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse. The hypotenuse is always the longest side of a right triangle, opposite the right angle.
  • Formula: For an angle \( \theta \) in a right triangle, \( \sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}} \).
  • Unit Circle: On the unit circle, the sine of an angle is the y-coordinate of the corresponding point.
  • Range: The sine function has values that range from -1 to 1.
Sine is foundational for understanding wave-like patterns and oscillations. This is why it鈥檚 crucial in fields like physics and engineering.
Cosine
Cosine, abbreviated as "cos," is closely related to sine as it also deals with sides of a right triangle, but instead it is the ratio of the adjacent side to the hypotenuse.
  • Formula: For an angle \( \theta \), \( \cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}} \).
  • Unit Circle: On the unit circle, the cosine of an angle is the x-coordinate of the point on the circle.
  • Range: Similar to sine, cosine values too range from -1 to 1.
Cosine is particularly important for solving problems involving diagonal distances and angles in geometry. It鈥檚 similarly pivotal in the analysis of periodic functions like sound waves.
Tangent
The tangent function, often abbreviated as "tan," is another primary trigonometric function. The tangent of an angle is the ratio of the sine of that angle to the cosine of that same angle.
  • Identity: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
  • Relation to Triangle: In a right triangle, tangent is the ratio of the opposite side to the adjacent side.
  • Value Range: Unlike sine and cosine, the range of the tangent function is all real numbers, since it can stretch to infinity as cosine approaches zero.
Understanding tangent is pivotal when dealing with slopes and angles of elevation or depression. It's common in various fields such as architecture, astronomy, and navigation.

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Most popular questions from this chapter

A bridge is to be built across a small lake from a gazebo to a dock (see figure). The bearing from the gazebo to the dock is \(\mathrm{S} 41^{\circ} \mathrm{W}\). From a tree 100 meters from the gazebo, the bearings to the gazebo and the dock are \(\mathrm{S} 74^{\circ} \mathrm{E}\) and \(\mathrm{S} 28^{\circ} \mathrm{E}\), respectively. Find the distance from the gazebo to the dock.

Fill in the blank. \(\sin (u+v)=\)_____

A Ferris wheel is built such that the height \(h\) (in feet) above ground of a seat on the wheel at time \(t\) (in minutes) can be modeled by \(h(t)=53+50 \sin \left(\frac{\pi}{16} t-\frac{\pi}{2}\right)\) The wheel makes one revolution every 32 seconds. The ride begins when \(t=0\). (a) During the first 32 seconds of the ride, when will a person on the Ferris wheel be 53 feet above ground? (b) When will a person be at the top of the Ferris wheel for the first time during the ride? If the ride lasts 160 seconds, how many times will a person be at the top of the ride, and at what times?

The length \(s\) of a shadow cast by a vertical gnomon (a device used to tell time) of height \(h\) when the angle of the sun above the horizon is \(\theta\) (see figure) can be modeled by the equation \(s=\frac{h \sin \left(90^{\circ}-\theta\right)}{\sin \theta}\) (a) Verify that the equation for \(s\) is equal to \(h \cot \theta\). (b) Use a graphing utility to complete the table. Let \(h=5\) feet. $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \theta & 15^{\circ} & 30^{\circ} & 45^{\circ} & 60^{\circ} & 75^{\circ} & 90^{\circ} \\ \hline s & & & & & & \\ \hline \end{array} $$ (c) Use your table from part (b) to determine the angles of the sun that result in the maximum and minimum lengths of the shadow. (d) Based on your results from part (c), what time of day do you think it is when the angle of the sun above the horizon is \(90^{\circ} ?\)

Let \(R\) and \(r\) be the radii of the circumscribed and inscribed circles of a triangle \(A B C\), respectively (see figure), and let \(s=\frac{a+b+c}{2}\). (a) Prove that \(2 R=\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). (b) Prove that \(r=\sqrt{\frac{(s-a)(s-b)(s-c)}{s}}\).
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