Ch. 1 - Trigonometric Functions

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 1 - Trigonometric Functions

Problem 35

Chapter 2, Problem 35

Find the measure of the smaller angle formed by the hands of a clock at the following times. 3:15

Verified step by step guidance

Understand that the problem asks for the smaller angle between the hour and minute hands of a clock at 3:15.

Calculate the position of the minute hand: since the minute hand moves 360 degrees in 60 minutes, its angle from the 12 o'clock position is given by \(\text{Minute Angle} = 6 \times \text{minutes}\). For 15 minutes, this is \(6 \times 15\) degrees.

Calculate the position of the hour hand: the hour hand moves 360 degrees in 12 hours, so it moves 30 degrees per hour. Additionally, it moves 0.5 degrees per minute. The formula for the hour hand angle is \(\text{Hour Angle} = 30 \times \text{hours} + 0.5 \times \text{minutes}\). For 3:15, substitute hours = 3 and minutes = 15.

Find the difference between the two angles calculated: \(\text{Angle Difference} = |\text{Hour Angle} - \text{Minute Angle}|\).

Since the clock is circular, the smaller angle between the hands is the minimum of \(\text{Angle Difference}\) and \(360 - \text{Angle Difference}\). This gives the measure of the smaller angle formed at 3:15.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Angle Measurement of Clock Hands

The position of clock hands can be translated into angles measured from the 12 o'clock position. The minute hand moves 6 degrees per minute (360°/60), while the hour hand moves 0.5 degrees per minute (30° per hour). Understanding these rates is essential to calculate the exact angles at any given time.

Calculating the Angle Between Two Lines

The angle between two clock hands is found by taking the absolute difference of their individual angles from the 12 o'clock position. Since the clock is circular, if this difference exceeds 180 degrees, the smaller angle is found by subtracting it from 360 degrees.

Time to Angle Conversion for the Hour Hand

Unlike the minute hand, the hour hand moves continuously as time passes. At 3:15, the hour hand is not exactly at 3 but has moved a quarter of the way towards 4. This continuous movement is calculated by adding 0.5 degrees per minute to the base hour angle.

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Related Practice

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Concept Check Classify each triangle as acute, right, or obtuse. Also classify each as equilateral, isosceles, or scalene. See the discussion following Example 2.

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Textbook Question

Determine the signs of the trigonometric functions of an angle in standard position with the given measure. See Example 2.

―15°

Textbook Question

Concept Check Suppose that the point (x, y) is in the indicated quadrant. Determine whether the given ratio is positive or negative. Recall that r = √(x² + y²) .(Hint: Drawing a sketch may help.) IV , x/y

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Textbook Question

Find the measure of the smaller angle formed by the hands of a clock at the following times.

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Textbook Question

An equation of the terminal side of an angle θ in standard position is given with a restriction on x. Sketch the least positive such angle θ , and find the values of the six trigonometric functions of θ . 12x + 5y = 0 , x ≥ 0 .

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Textbook Question

Determine the signs of the trigonometric functions of an angle in standard position with the given measure. See Example 2.

―345°