Trigonometry Basics Quiz

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Questions and Answers

What does the word 'trigonometry' derive from?

Latin words for triangle and measure (correct)
Greek words for angle and distance
Latin words for angle and measure
Greek words for triangle and ratio

Which mathematical branches does trigonometry interrelate with?

Calculus, Algebra, and Geometry (correct)
Geography, Physics, and Algebra
Statistics, Algebra, and Geometry
Physics, Calculus, and Logic

How are angles measured in trigonometry?

In meters and centimeters
In degrees and radians (correct)
In centimeters and kilograms
In radians and proportions

What is the purpose of trigonometric functions?

To find relationships between lengths and angles in triangles (A) Signup and view all the answers

What is NOT a fundamental identity in trigonometry?

Quadratic Formula (D) Signup and view all the answers

Which of the following describes a key feature of the sine function?

A periodic graph with a maximum and minimum value (C) Signup and view all the answers

What shape do angles in a triangle form?

An initial side and a terminal side (D) Signup and view all the answers

Which formula would be used to find a missing side length given an angle measure in a right triangle?

Pythagorean Theorem (D) Signup and view all the answers

What is the value of sin 30°?

0.5 (A) Signup and view all the answers

What is the value of cos 45°?

0.7071 (A) Signup and view all the answers

What is the correct ratio for tan 60°?

√3 (A) Signup and view all the answers

How do you calculate cos 55° using a calculator?

Input 55 into the cosine function and press enter. (B) Signup and view all the answers

What is the result of tan 0°?

0 (D) Signup and view all the answers

If given an angle measure and a side length, how can you find other side lengths?

Formulate an equation using trig functions. (A) Signup and view all the answers

Which of the following ratios corresponds to the special triangle with a 45° angle?

1:1:√2 (D) Signup and view all the answers

What steps should you take to ensure your TI Graphing calculator is set correctly for trig functions?

Set to degrees, then press mode. (A) Signup and view all the answers

What is the first step in applying the Law of Sines to find the length of AB?

Eliminate fractions by multiplying by a common denominator (A) Signup and view all the answers

When can the Law of Cosines be used?

When all three sides and one angle are known (D) Signup and view all the answers

What is the formula for the Law of Cosines?

$c^2 = a^2 + b^2 - 2ab ext{ cos } C$ (C) Signup and view all the answers

What value did the application of the Law of Sines yield for AB in the first example?

7 (D) Signup and view all the answers

What is the relationship used to find AB when applying the Law of Sines?

$sin 28° = rac{AB}{15}$ (D) Signup and view all the answers

What is the sum formula for sine?

sin(𝛼 + 𝛽) = sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽 (B) Signup and view all the answers

What is the formula for cosine of the difference of two angles?

cos(𝛼 − 𝛽) = cos 𝛼 cos 𝛽 + sin 𝛼 sin 𝛽 (D) Signup and view all the answers

What is the correct breakdown of cos 105° using sum formulas?

cos 105° = cos(60° + 45°) (D) Signup and view all the answers

What is the formula for tangent of the difference of two angles?

tan(𝛼 − 𝛽) = rac{tan 𝛼 - tan 𝛽}{1 + tan 𝛼 tan 𝛽} (C) Signup and view all the answers

What identity is used to find cos 30°?

cos 30° = sin(90° − 30°) (C) Signup and view all the answers

How is sin 15° calculated using angle difference formulas?

sin 15° = sin(45° – 30°) (D) Signup and view all the answers

What is the result of the formula tan(𝛼 + 𝛽) = rac{tan 𝛼 + tan 𝛽}{1 - tan 𝛼 tan 𝛽}?

The formula calculates the tangent of a sum of two angles. (D) Signup and view all the answers

What is true about the cofunction identity for sin?

sin(90° − 𝜃) = cos 𝜃 (B) Signup and view all the answers

What is the correct value of cos 15° according to the half angle formula?

±√3/2 (D) Signup and view all the answers

Which quadrant is the angle 𝜃 in if cos 𝜃 = -4/5 and 90° < 𝜃 < 180°?

Second quadrant (A) Signup and view all the answers

What is the sine value of 𝜃 if cos 𝜃 = -4/5 and 90° < 𝜃 < 180°?

3/5 (B) Signup and view all the answers

Which product-to-sum formula corresponds to sin A cos B?

sin A cos B = (1/2)(sin(A + B) + sin(A − B)) (B) Signup and view all the answers

What is the result when using the product-to-sum formula for sin 75° sin 15°?

1/4 (D) Signup and view all the answers

What formula would you use to find missing sides in oblique triangles?

Law of Sines (D) Signup and view all the answers

How can you transform sin A − sin B using the sum-to-product formulas?

2 cos((A + B)/2) sin((A - B)/2) (A) Signup and view all the answers

What is the expression for cos A + cos B in its sum-to-product form?

2 cos((A + B)/2) cos((A - B)/2) (D) Signup and view all the answers

What is the complete formula of the Law of Sines?

sin A/sin B = a/b (C) Signup and view all the answers

What is the result of sin 70° − sin 30° when transformed into a product?

2 cos(50°) sin(20°) (B) Signup and view all the answers

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Study Notes

Trigonometry Overview

Trigonometry examines relationships between triangle sides and angles.
The term derives from Greek words "trigonon" (triangle) and "metron" (measure).
It intertwines with geometry, algebra, and calculus.

Angle Measurement

Angles are measured in degrees and radians.
The unit circle is a crucial concept in trigonometry.
Special triangles (30°, 45°, 60°) have fixed side length ratios for quicker calculations.

Trigonometric Functions

Defined trig ratios: sin, cos, and tan relate angles to triangle sides.
Trig functions can indicate values given an angle or reveal pitch from lengths.
Essential for finding unknown sides or angles in right triangles.

Fundamental Identities

Fundamental identities provide foundational relationships among trig functions.
Sum and Difference formulas help to compute trig functions of compound angles.
The Double Angle, Half Angle, and Product-to-Sum formulas facilitate further value manipulations.

Law of Sines and Cosines

The Law of Sines relates angles to opposite sides:
- sin(A)/a = sin(B)/b = sin(C)/c.
The Law of Cosines connects all sides and one angle:
- c² = a² + b² - 2ab * cos(C).

Graphs of Trig Functions

Sine and cosine graphs exhibit periodic characteristics.
Key features include amplitude, period, and phase shift.
The graph of the tangent function also has unique properties and periodicity.

Special Triangle Ratios

For a 30° angle:
- sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = √3/3.
For a 45° angle:
- sin(45°) = √2/2, cos(45°) = √2/2, tan(45°) = 1.
For a 60° angle:
- sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3.

Calculators in Trigonometry

Use graphing calculators to find trig function values.
Set calculators to degrees or radians according to the problem requirements.

Advanced Trig Formulas

Half Angle formula assists in simplifying trigonometric expressions.
Product-to-Sum and Sum-to-Product formulas convert between forms for easier calculations.

Cofunction Identities

Relate sine, cosine, tangent, secant, cosecant, and cotangent through:
- cos(90° - θ) = sin(θ) and similar relationships for other functions.

Practical Applications

Trigonometry is vital in physics, engineering, and architecture.
Skills in identifying various angles and sides increase problem-solving efficiency in right and non-right triangles.

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