Trigonometry Chapter 8 Exercises

Study Notes

Triangle ABC Properties

Triangle ABC is right-angled at B with sides AB = 24 cm and BC = 7 cm.
To calculate trigonometric values for angles A and C:
- ( \sin A ) and ( \cos A ) are determined based on the sides opposite and adjacent to angle A.
- ( \sin C ) and ( \cos C ) are calculated similarly for angle C.

Tangent and Cotangent Relationships

Use of the tangent and cotangent functions in various identities:
- ( \tan P - \cot R ) can reflect the relationship between angles and the sides of a figure.

Additional Trigonometric Calculations

When ( \sin A = \frac{3}{4} ), it's essential to calculate:
- ( \cos A ) using the Pythagorean identity.
- ( \tan A ) as the ratio of ( \sin A ) and ( \cos A ).

Cosecant and Secant Computations

If ( 15 \cot A = 8 ):
- Deriving ( \sin A ) and ( \sec A ) from the meaning of cotangent in right-angled triangles.

Comprehensive Trigonometric Ratios

For ( \sec \theta = \frac{13}{12} ):
- Calculation of ( \sin \theta, \cos \theta, \tan \theta, \cot \theta, \text{ and } \csc \theta ) using the trigonometric identities.

Angle Equality Proof

If ( \cos A = \cos B ) with ( \angle A ) and ( \angle B ) acute:
- The implication is ( \angle A = \angle B ), showcasing the properties of cosine functions for acute angles.

Evaluating Trigonometric Expressions

Given ( \cot \theta = \frac{8}{7} ):
- Evaluations for ( (1 + \sin \theta)(1 - \sin \theta) / (1 + \cos \theta)(1 - \cos \theta) ) and ( \cot^2 \theta ).

Verification of Trigonometric Identities

Check if ( \frac{1 - \tan^2 A}{1 + \tan^2 A} = \cos^2 A - \sin^2 A ) when ( 3 \cot A = 4 ).

Triangle PQR Characteristics

Triangle PQR is right-angled at Q with the lengths ( PR + QR = 25 \text{ cm} ) and ( PQ = 5 \text{ cm} ).
- Find values for ( \sin P, \cos P, \text{ and } \tan P ).

True or False Statements

Evaluating statements on trigonometric values:
- ( \tan A < 1 ): False, depends on the angle.
- ( \sec A = \frac{12}{5} ): True for certain angles.
- Abbreviations: ( \cos A ) is not the same as cosecant.
- ( \cot A ) is not represented as ( \text{cot} \cdot A ).
- ( \sin \theta = \frac{4}{3} ): False, sine values are limited to [-1, 1].

Trigonometric Ratios of Specific Angles

Trigonometric ratios for angles ( 0°, 30°, 45°, 60°, 90° ) need to be memorized, as they have specific values:
- Useful for geometry and further trigonometric applications.

Description

Test your knowledge of trigonometric functions with these exercises designed from Chapter 8. Solve problems related to right-angled triangles, trigonometric ratios, and identities. Perfect for math students looking to reinforce their understanding of the subject.