# 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 = √(1^{2} + 1^{2}).

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 45^{0}.

#### 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) ≈ 61.88^{0}.
Therefore the angle A

is approximately 61.88^{0}.

### 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 60^{0}. Sin of 60^{0} equals approximately 0.866.
Asin (arcsine) of 0.866 is approximately 60^{0}.

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 ≈ 61.88^{0}

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.

### Post Your Score Online

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.