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Solution - Trigonometry

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Trigonometry

Step-by-step explanation

1. Solve trigonometry

The period of trigonometric functions is 360 degrees.

tan(585°)=tan(585-360°)

Subtracting one integer from another.

tan(585-360°)=tan(225°)

Reflecting the number with respect to 360 degrees.

tan(225°)=tan(360-135°)

The period of trigonometric functions is 360 degrees.

tan(360-135°)=tan(360-135-360°)

Removing or simplifying the same numbers on the top and bottom of a fraction.

tan(360-135-360°)=tan(-135°)

The tangent of an angle is equal to the sine of the angle divided by the cosine of the angle.

tan(-135°)=sin(-135°)cos(-135°)

Computing the sine of a negative angle.

sin(-135°)cos(-135°)=-sin(135°)cos(-135°)

Computing the cosine of a negative angle.

-sin(135°)cos(-135°)=-sin(135°)cos(135°)

Placing the minus sign in front of a fraction.

-sin(135°)cos(135°)=-sin(135°)cos(135°)

The tangent of an angle is equal to the sine of the angle divided by the cosine of the angle.

-sin(135°)cos(135°)=-tan(135°)

Reflecting the number with respect to 360 degrees.

-tan(135°)=-tan(180-45°)

The tangent of an angle is equal to the sine of the angle divided by the cosine of the angle.

tan(180-45°)=sin(180-45°)cos(180-45°)

Reflecting the sine function with respect to 180 degrees.

sin(180-45°)cos(180-45°)=sin(45°)cos(180-45°)

Reflecting the cosine function with respect to 180 degrees.

sin(45°)cos(180-45°)=sin(45°)-cos(45°)

Placing the minus sign in front of a fraction.

sin(45°)-cos(45°)=-sin(45°)cos(45°)

The tangent of an angle is equal to the sine of the angle divided by the cosine of the angle.

-sin(45°)cos(45°)=-tan(45°)

The tangent of an angle is equal to the sine of the angle divided by the cosine of the angle.

tan(45°)=sin(45°)cos(45°)

Computing the sine of 45 degrees.

sin(45°)cos(45°)=22cos(45°)

Computing the cosine of 45 degrees.

22cos(45°)=2222

Dividing the same numbers.

2222=1

Why learn this

Trigonometry is a branch of mathematics that deals with the relationships between angles and sides of triangles. It may sound complex, but trigonometry is actually quite useful in many real-life situations. Let's dive in and explore why learning trigonometry is important and how it relates to everyday life.

Understanding Angles:
Trigonometry helps us understand angles and their measurements. Imagine you're planning a picnic with your friends, and you want to find the perfect spot to set up your picnic blanket. You can use trigonometry to determine the angle of the sun and find a shady spot to avoid direct sunlight.

Navigation and Distance:
Trigonometry is crucial for navigation and calculating distances. When you're using a GPS or a map app on your phone to find the shortest route to a destination, it's actually using trigonometric functions to calculate distances and angles between different points.

Building and Construction:
Trigonometry plays a vital role in architecture and construction. Architects and engineers use trigonometric concepts to design structures, determine the height of buildings, calculate angles for roofs, and ensure stability and safety in construction projects.

Astronomy and Celestial Navigation:
Trigonometry has long been used in astronomy and celestial navigation. Ancient astronomers used trigonometric principles to measure distances between stars and planets. Today, trigonometry helps scientists understand the movement of celestial bodies and even explore space.

Sports and Games:
Trigonometry can be found in various sports and games. For example, if you enjoy playing baseball or cricket, understanding the angles and trajectories of the ball can help you improve your aim. Trigonometry is also used in activities like billiards, golf, and even video games to calculate angles and predict movements.

Sound and Waves:
Trigonometry is essential in the study of sound and waves. Musicians and audio engineers use trigonometric concepts to understand waveforms, harmonics, and frequencies. It helps in tuning musical instruments and designing sound systems.

These are just a few examples of how trigonometry is relevant to our daily lives. By learning trigonometry, you'll develop problem-solving skills, enhance your spatial reasoning, and gain a deeper understanding of the world around you. So, embrace trigonometry as a valuable tool that can be applied in various fields and make your everyday life more exciting and meaningful!

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