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# Help Solve for Two Angles

## Trig Triumph Game: Level 5 Help

### Show Minutes and Seconds as Digits after the Decimal Point.

Level 5 Find Two Angles requires two answers for each question. First use the calculator to determine Angle α then tap the button labeled Answer A. Your answer then displays on the button. Second calculate Angle β, then tap the button labeled Answer B. If you're not happy with your answers, then simply recalculate and tap the Answer buttons. When you're satisfied with both answers, tap the Final Answer button.

### Calculate Two Angles

This Web page demonstrates how to solve for two angles, given the lengths of two sides of a right triangle,

### Inverse Trigonometric Functions

The Ratios level demonstrated the trigonometric ratios sine, cosine, tangent, cosecant, secant, and cotangent. For each trigonometric function there's an inverse trigonometric function. Inverse trigonmetric functions receive an input and return an output. Input the Trigonmetric Ratios from level two. Inverse trigonometric functions output an angle in degrees. In other words provide a trigonometric ratio to an inverse trigonometric function, to find the associated angle. ### Find Two Angles Given Two Sides of a Right Triangle

#### Example 1

Given a right triangle where the lengths of the vertical and horizontal sides are both 1. The hypotenuse is 1.414.

We know the hypotenuse because:

```c = √(12 + 12)..
1.414 ≈ √(1 + 1)
```

The sin of 1.414 would be 1/1.414 ≈ 0.707

The arcsin(0.707) ≈ 45.

Therefore both angles are 450.

#### Example 2

Given a triangle with hypotenuse 17" and side a is 15". The Sin of A is 15/17. 15/17 ≈ 0.882. asin(0.882) ≈ 61.880. Therefore the angle A is approximately 61.880.

### Compositional Inverse

The terms compositional inverse and multiplicative inverse represent different operations. It's easy to become confused regarding the term inverse. A multiplicative inverse is the reciprocal of a fraction. For example, the multiplicative inverse of 1/2 is 2/1.

The inverse trigonometric functions are compositional inverses not multiplicative inverses. A compositional or functional inverse, reverses the effect of another function. For example, assume angle A equals 600. Sin of 600 equals approximately 0.866. Asin (arcsine) of 0.866 is approximately 600.

In simpler terms, if the sin of y = x then the arcsin of x = y.

Look at Example 2 above, we can see that the Sin of 61.88 ≈ 0.882. The arcsin of 0.882 ≈ 61.880

Inverse trigonometric function output is restricted to provide only one possible solution.

### Notation For the Game

The inverse trigonometric functions may be represented with different notations, depending on the text or instructor. For example the inverse function for Sin may be noted as arcsin, Sin-1, INV Sin, or asin. The Trig Triumph game uses the notation listed in the table above for the set of inverse trigonometric functions. The inverse trigonometric functions include asin, acos, atan, acsc, asec, and acot.

Select the Post Score button to post your score online. See how your score compares to others in the Trig Triumph Hall of Fame.

#### Rounding Errors

Trig Triumph rounds the final answer to the nearest thousandth (3 digits after the decimal point). The Trig Triumph game allows for small rounding differences. The game rounds to the nearest three digits, and displays the output, after you tap the Final Answer button.

Often rounding each step in a series introduces rounding errors. The final result may be too high, when more than one step in the series rounds up. The final result may be too low, when more than one step in the series rounds down. Therefore it's best to avoid rounding until the final answer. Let Trig Triumph then round the final answer.

### Reset Level Five

Select the Reset button at the bottom of the page, to begin this level again. Reset allows you to play the level over again and increase your score. Reset assigns zero to your score for this level, then displays the first question for this level.