### Identifying True Statements in a Scalene TriangleConsider the right-angled triangle $ \Delta CAB $ shown in the diagram. For this triangle:- $ \angle CAB $ is represented by $ \alpha $- $ \angle CBA $ (the right angle)- $ \angle ADB $ is represented by $ \beta $The triangle has sides labeled as follows:- Side $ a $ opposite angle $ \alpha $- Side $ b $ adjacent to angle $ \alpha $- Hypotenuse $ c $### Problem StatementIdentify ALL of the following statements that are true in the given $ \Delta CAB $ where $ \alpha + \beta = 90^\circ $: (1 point)- **A.** $ \sin(\beta) = \cos(\alpha) $- **B.** $ \tan(\beta) = \tan(\alpha) $- **C.** $ \cos(\beta) = \sin(\alpha) $- **D.** $ \cos(\beta) = \sin(90^\circ - \alpha) $**Answer Options:**- ☐ A- ☐ B- ☐ C- ☐ D### Diagram ExplanationThe diagram provided is a right-angled triangle labeled $ \Delta CAB $:- $ \angle CBA = 90^\circ $- $ \angle CAB = \alpha $- $ \angle ADB = \beta $- Side opposite $ \alpha $ is $ a $- Side adjacent to $ \alpha $ is $ b $- Hypotenuse is $ c $### Analysis of Statements- **Statement A:** $ \sin(\beta) = \cos(\alpha) $ - By trigonometric identity in a right triangle, this is true because $ \beta = 90^\circ - \alpha $. Therefore, $ \sin(\beta) = \sin(90^\circ - \alpha) = \cos(\alpha) $.- **Statement B:** $ \tan(\beta) = \tan(\alpha) $ - This is false. $ \tan(\beta) = \tan(90^\circ - \alpha) = \cot(\alpha) $. Therefore, \( \tan(\beta) \neq \tan(\alpha)

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Description: ### Identifying True Statements in a Scalene TriangleConsider the right-angled triangle $ \Delta CAB $ shown in the diagram. For this triangle:- $ \angle CAB $ is represented by $ \alpha $- $ \angle CBA $ (the right angle)- $ \angle ADB $

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Rectangular Snip Identify ALL of the following statements that are true in the diagram if ACAB is a scalene triangle. (1 point) в O A. sin(B) = cos(a) O B. tan(B) = tan(a) %3D O C. cos (B) = sin (a) O D. cos(B) = sin(90 – a) A b A

Rectangular Snip Identify ALL of the following statements that are true in the diagram if ACAB is a scalene triangle. (1 point) в O A. sin(B) = cos(a) O B. tan(B) = tan(a) %3D O C. cos (B) = sin (a) O D. cos(B) = sin(90 – a) A b A

Elementary Geometry For College Students, 7e

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Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

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Concept explainers

Fundamentals of Trigonometric Identities

Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.

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Question

### Identifying True Statements in a Scalene Triangle

Consider the right-angled triangle $ \Delta CAB $ shown in the diagram. For this triangle:

- $ \angle CAB $ is represented by $ \alpha $
- $ \angle CBA $ (the right angle)
- $ \angle ADB $ is represented by $ \beta $

The triangle has sides labeled as follows:
- Side $ a $ opposite angle $ \alpha $
- Side $ b $ adjacent to angle $ \alpha $
- Hypotenuse $ c $

### Problem Statement

Identify ALL of the following statements that are true in the given $ \Delta CAB $ where $ \alpha + \beta = 90^\circ $: (1 point)

- **A.** $ \sin(\beta) = \cos(\alpha) $
- **B.** $ \tan(\beta) = \tan(\alpha) $
- **C.** $ \cos(\beta) = \sin(\alpha) $
- **D.** $ \cos(\beta) = \sin(90^\circ - \alpha) $

**Answer Options:**

- ☐ A
- ☐ B
- ☐ C
- ☐ D

### Diagram Explanation

The diagram provided is a right-angled triangle labeled $ \Delta CAB $:

- $ \angle CBA = 90^\circ $
- $ \angle CAB = \alpha $
- $ \angle ADB = \beta $
- Side opposite $ \alpha $ is $ a $
- Side adjacent to $ \alpha $ is $ b $
- Hypotenuse is $ c $

### Analysis of Statements

- **Statement A:** $ \sin(\beta) = \cos(\alpha) $
- By trigonometric identity in a right triangle, this is true because $ \beta = 90^\circ - \alpha $. Therefore, $ \sin(\beta) = \sin(90^\circ - \alpha) = \cos(\alpha) $.

- **Statement B:** $ \tan(\beta) = \tan(\alpha) $
- This is false. $ \tan(\beta) = \tan(90^\circ - \alpha) = \cot(\alpha) $. Therefore, \( \tan(\beta) \neq \tan(\alpha)

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