https://file.aiquickdraw.com/mathbot/file/17290020849319n0hwgkz.jpg

The problem asks to express trigonometric ratios with large angles into their corresponding acute angles using standard trigonometric identities. We can reduce each angle by subtracting multiples of 360° (since trigonometric functions are periodic with a period of 360°) to convert them into an equivalent angle within the first circle (0° to 360°), and then apply the necessary transformations to represent them in terms of acute angles.

Let's break this down for each part:

### a) \( \sin 528^\circ \)
To reduce:
\[
528^\circ - 360^\circ = 168^\circ
\]
Now, using \( \sin(180^\circ - \theta) = \sin(\theta) \):
\[
\sin(528^\circ) = \sin(168^\circ) = \sin(12^\circ)
\]

### b) \( \cos 634^\circ \)
To reduce:
\[
634^\circ - 360^\circ = 274^\circ
\]
Using \( \cos(360^\circ - \theta) = \cos(\theta) \):
\[
\cos(634^\circ) = \cos(274^\circ) = -\sin(4^\circ)
\]

### c) \( \csc 432^\circ \)
To reduce:
\[
432^\circ - 360^\circ = 72^\circ
\]
The cosecant function follows the same rule as sine. Therefore:
\[
\csc(432^\circ) = \csc(72^\circ)
\]

### d) \( \sec 567^\circ \)
To reduce:
\[
567^\circ - 360^\circ = 207^\circ
\]
Using \( \sec(180^\circ + \theta) = -\sec(\theta) \):
\[
\sec(567^\circ) = -\sec(27^\circ)
\]

### e) \( \tan 872^\circ \)
To reduce:
\[
872^\circ - 720^\circ = 152^\circ
\]
Using \( \tan(180^\circ - \theta) = -\tan(\theta) \):
\[
\tan(872^\circ) = -\tan(28^\circ)
\]

### f) \( \cos 1182^\circ \)
To reduce:
\[
1182^\circ - 1080^\circ = 102^\circ
\]
Using \( \cos(180^\circ - \theta) = -\cos(\theta) \):
\[
\cos(1182^\circ) = -\cos(78^\circ)
\]

### g) \( \tan 2100^\circ \)
To reduce:
\[
2100^\circ - 1980^\circ = 120^\circ
\]
Using \( \tan(180^\circ - \theta) = -\tan(\theta) \):
\[
\tan(2100^\circ) = -\tan(60^\circ)
\]

### h) \( \sin 1563^\circ \)
To reduce:
\[
1563^\circ - 1440^\circ = 123^\circ
\]
Using \( \sin(180^\circ - \theta) = \sin(\theta) \):
\[
\sin(1563^\circ) = \sin(57^\circ)
\]

### i) \( \sec 987^\circ \)
To reduce:
\[
987^\circ - 720^\circ = 267^\circ
\]
Using \( \sec(360^\circ - \theta) = \sec(\theta) \):
\[
\sec(987^\circ) = -\sec(87^\circ)
\]

### j) \( \tan 1123^\circ \)
To reduce:
\[
1123^\circ - 1080^\circ = 43^\circ
\]
Thus:
\[
\tan(1123^\circ) = \tan(43^\circ)
\]

### k) \( \sec 1567^\circ \)
To reduce:
\[
1567^\circ - 1440^\circ = 127^\circ
\]
Using \( \sec(180^\circ - \theta) = -\sec(\theta) \):
\[
\sec(1567^\circ) = -\sec(53^\circ)
\]

### l) \( \csc 1445^\circ \)
To reduce:
\[
1445^\circ - 1080^\circ = 365^\circ - 360^\circ = 5^\circ
\]
Thus:
\[
\csc(1445^\circ) = \csc(5^\circ)
\]

Would you like further details or explanations on any of these reductions? Here are five follow-up questions to consider:

1. How does the periodicity of trigonometric functions help in simplifying angles?
2. Why is it important to express trigonometric ratios in terms of acute angles?
3. Can you identify the quadrant for each angle after reducing them to the range \(0^\circ\) to \(360^\circ\)?
4. What is the significance of applying the co-function identities for \(180^\circ\) and \(360^\circ\)?
5. How can we use unit circle properties to understand the signs of trigonometric functions in different quadrants?

**Tip:** Always reduce angles greater than 360° by subtracting multiples of 360° to simplify calculations.

Nyatakan perbandingan trigonometri berikut ini ke dalam perbandingan trigonometri sudut lancip: sin 528°, cos 634°, cosec 432°, sec 567°, tan 872°, cos 1182°, tan 2100°, sin 1563°, sec 987°, tan 1123°, sec 1567°, cosec 1445°.

Learn how to simplify trigonometric ratios such as sin 528°, cos 634°, and sec 567° by reducing them to acute angles using periodicity and identities. This guide covers key trigonometric concepts and formulas, suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation