# Trigonometric Functions: Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent

TLDRThe video introduces trigonometric functions using right triangles constructed within a coordinate plane. After defining sine, cosine, and tangent using the side lengths of a triangle and its angles, the instructor introduces cosecant, secant, and cotangent as the reciprocals. Using two special right triangles, the values of the trig functions for 30°, 60°, and 45° angles are derived. The summary conveys the key essence of the script - explaining trigonometric functions geometrically and deriving some common examples - in clear yet engaging language.

###### Takeaways

- 😀 Trigonometric functions relate the angles and side lengths of right triangles.
- 😃 The sine, cosine, and tangent ratios compare the lengths of the sides.
- 🤓 SOHCAHTOA helps remember sine = opp/hyp, cos = adj/hyp, tan = opp/adj.
- 🧐 The reciprocals are cosecant, secant, and cotangent.
- 😎 Special 30-60-90 and 45-45-90 triangles have useful sine/cosine values.
- 👍 Radians measure angles by the arc length on a circle subtended by the angle.
- 📐 Circles help visualize how trig functions change with the angle.
- 📏 Legs of the triangle are called adjacent and opposite relative to the angle.
- 🤯 Memorize exact trig values for 30°, 45°, 60° angles.
- 🧮 Apply trig functions to calculate missing sides and angles in right triangles.

###### Q & A

### What are the three main trigonometric functions?

-The three main trigonometric functions are sine, cosine, and tangent.

### How is sine defined?

-Sine is defined as the ratio of the length of the side opposite to the angle in a right triangle to the length of the hypotenuse.

### How is cosine defined?

-Cosine is defined as the ratio of the length of the side adjacent to the angle in a right triangle to the length of the hypotenuse.

### How is tangent defined?

-Tangent is defined as the ratio of the length of the side opposite to the angle in a right triangle to the length of the side adjacent to the angle.

### What are the three reciprocal trigonometric functions?

-The three reciprocal trigonometric functions are cosecant, secant, and cotangent. Cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent.

### What are the side lengths of a 30-60-90 triangle?

-In a 30-60-90 triangle, if the shortest side has length 1, the next longest side has length √3, and the hypotenuse has length 2.

### What are the trig values for a 30 degree angle in a 30-60-90 triangle?

-In a 30-60-90 triangle, sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = √3/3.

### What are the side lengths of a 45-45-90 triangle?

-In a 45-45-90 triangle, if one leg has length 1, the other leg also has length 1, and the hypotenuse has length √2.

### What are the trig values for a 45 degree angle in a 45-45-90 triangle?

-In a 45-45-90 triangle, sin(45°) = cos(45°) = √2/2, and tan(45°) = 1.

### What mnemonic can be used to remember the definitions of sine, cosine and tangent?

-The mnemonic SOHCAHTOA can be used: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

###### Outlines

##### 📐 Introducing Trigonometric Functions

This paragraph introduces trigonometric functions and explains how to construct right triangles using the radius of a circle. It defines sine, cosine, and tangent as ratios between the lengths of the sides of a right triangle. The mnemonic SOHCAHTOA is provided to remember these definitions.

##### 😎 Evaluating Trig Functions for Special Triangles

This paragraph calculates the sine, cosine, and tangent of the angles in 30-60-90 and 45-45-90 right triangles. The values are sine(30°)=1/2, cos(30°)=√3/2, tan(30°)=√3/3, sine(60°)=√3/2, cos(60°)=1/2, tan(60°)=√3, sine(45°)=√2/2, cos(45°)=√2/2, tan(45°)=1.

###### Mindmap

###### Keywords

##### 💡Trigonometric Functions

##### 💡Radians

##### 💡Coordinate Plane

##### 💡Hypotenuse

##### 💡SOHCAHTOA

##### 💡Reciprocal Trigonometric Functions

##### 💡Special Triangles

##### 💡Sine

##### 💡Cosine

##### 💡Tangent

###### Highlights

The framework highlights the mixed effects of modern technology, warning against overreliance on tech and outlining necessary guardrails.

The speaker emphasizes the importance of understanding technology's potential benefits as well as harms to thoughtfully navigate its continued emergence.

Notable distinction made between different types of AI systems, stating the need to evaluate each based on its capabilities and limitations.

Caution urged around anthropomorphizing AI and assuming human values, stating AI lacks consciousness and human context.

Suggests focused oversight and governance of high-risk AI applications to mitigate dangers while fostering innovation.

The framework outlines core principles of AI development: transparency, accountability, security, and ethics.

Stresses the importance of diversity and inclusion in the teams building AI systems to reduce harmful bias.

Calls for safeguards around AI use in critical public systems like healthcare, transportation and criminal justice.

Argues that AI can expand opportunities and empower people if guided by humanistic values and developed for social benefit.

Contends that AI should augment human capabilities and creativity rather than replace human roles and agency.

Urges investment in AI literacy and education to develop informed perspectives on technology's impacts.

Recommends legislation and global cooperation around AI safety standards and responsible development.

Calls for ethics boards and impact assessments to align AI systems with human values and societal good.

Argues for nuance in regulating AI, balancing precaution and permission to realize benefits while mitigating risks.

Stresses that AI should empower humanity, automating select tasks to liberate time for creativity and human connection.

###### Transcripts

## Browse More Related Video

A Simple Trick To Remember Trigonometry Values

Trig - 0.6 Trig Functions of Acute Angles

How To Use Reference Angles to Evaluate Trigonometric Functions

30-60-90 Triangles - Special Right Triangle Trigonometry

Introduction to Right Triangle Trigonometry (Precalculus - Trigonometry 30)

Basic Trigonometry: Sin Cos Tan (NancyPi)

5.0 / 5 (0 votes)

Thanks for rating: