# Solve Trig. Problem without a calculator

• morr485
In summary, the conversation discusses the relationship between sec^2(x) and tan^2(x), and how they can be used to solve equations involving trigonometric functions. By manipulating the given equations and using identities, one can solve for tan(x) and prove the relationship between these two functions.
morr485
1. sec^2(x) = (1 + sqrt3) - (1 - sqrt3)*tan (x)

2.sec^2(x) = tan^2(x) +1

3. tan^2(x) + tan(x) + (1 - sqrt3)*tan(x) - sqrt3 = 0
tan^(x)[tan(x) +1 +1 - sqrt3) -sqrt(3) = 0
tan^2(x) + (2 - sqrt3) - sqrt(3) = 0

morr485 said:
1. sec^2(x) = (1 + sqrt3) - (1 - sqrt3)*tan (x)

2.sec^2(x) = tan^2(x) +1

3. tan^2(x) + tan(x) + (1 - sqrt3)*tan(x) - sqrt3 = 0
tan^(x)[tan(x) +1 +1 - sqrt3) -sqrt(3) = 0
tan^2(x) + (2 - sqrt3) - sqrt(3) = 0

sec2(x) - 1 + (1 - sqrt(3))tan(x) - sqrt(3) = 0

Replace sec2(x) - 1 with tan2(x) and you will have a quadratic equation in tan(x), which you can solve by factoring (maybe) or by use of the quadratic formula.

For #2, sec^2 will equal 1/cos^2, and tan^2 will equal sin^2/cos^2. If you have learned the Pythagorean Identity (sin^2+cos^2=1) you can isolate cos^2 and solve the left side, which will leave you with 1/1-sin^2. Solving the right side, tan^2 will equal sin^2/cos^2, and you can make 1 equal to sin^2/cos^2, which would equal 2sin^2/cos^2. You can solve this equation to prove the left side.

## 1. How can I solve trigonometry problems without using a calculator?

There are several methods for solving trigonometry problems without a calculator, including using trigonometric identities, special triangles, and the unit circle. It is important to understand the basic trigonometric functions and their properties in order to solve problems without a calculator.

## 2. What are some common trigonometric identities that can be used to solve problems?

Some common trigonometric identities include the Pythagorean identities, sum and difference identities, double angle identities, and half angle identities. These identities can be used to simplify trigonometric expressions and equations.

## 3. How can I use special triangles to solve trigonometry problems?

Special triangles, such as the 30-60-90 and 45-45-90 triangles, have specific ratios between their sides that can be used to solve trigonometric problems. By memorizing these ratios and understanding how they relate to the trigonometric functions, you can solve problems without a calculator.

## 4. What is the unit circle and how can it be used to solve trigonometry problems?

The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. It is often used in trigonometry to understand the values of trigonometric functions at different angles. By memorizing the coordinates of key points on the unit circle, you can solve trigonometry problems without a calculator.

## 5. Are there any tips for solving trigonometry problems without a calculator?

One helpful tip is to practice and become familiar with the trigonometric functions and their properties. It can also be helpful to draw diagrams and label the given information in a problem. Additionally, memorizing the values of common angles, such as 0, 30, 45, 60, and 90 degrees, can make solving problems easier.

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