Question

If cos(90° - theta ) sin theta + cos theta sin(90° - theta ) is_​

Answer 1

Answer:

Let's start with a right triangle. Notice how the acute angles are complementary, sum to 90^\circ  

∘

degrees.

[Help! Please break this down for me.]

Now here's the cool part. See how the sine of one acute angle

 

describes the \blueD{\text{exact same ratio}}exact same ratiostart color #11accd, start text, e, x, a, c, t, space, s, a, m, e, space, r, a, t, i, o, end text, end color #11accd as the cosine of the other acute angle?

 

 

 

Incredible! Both functions, \sin(\theta)sin(θ)sine, left parenthesis, theta, right parenthesis and \cos(90^\circ-\theta)cos(90  

∘

−θ)cosine, left parenthesis, 90, degrees, minus, theta, right parenthesis, give the exact same side ratio in a right triangle.

And we're done! We've shown that \sin(\theta) = \cos(90^\circ-\theta)sin(θ)=cos(90  

∘

−θ)sine, left parenthesis, theta, right parenthesis, equals, cosine, left parenthesis, 90, degrees, minus, theta, right parenthesis.

In other words, the sine of an angle equals the cosine of its complement.

Well, technically we've only shown this for angles between 0^\circ  

∘

degrees and 90^\circ  

∘

degrees. To make our proof work for all angles, we'd need to move beyond right triangle trigonometry into the world of unit circle trigonometry, but that's a task for another time.

Cofunctions

You may have noticed that the words sine and cosine sound similar. That's because they're cofunctions! The way cofunctions work is exactly what you saw above. In general, if fff and ggg are cofunctions, then

f(\theta) = g(90^\circ-\theta)f(θ)=g(90  

∘

−θ)f, left parenthesis, theta, right parenthesis, equals, g, left parenthesis, 90, degrees, minus, theta, right parenthesis

and

g(\theta) = f(90^\circ-\theta)g(θ)=f(90  

∘

−θ)g, left parenthesis, theta, right parenthesis, equals, f, left parenthesis, 90, degrees, minus, theta, right parenthesis.

Here is a full list of the basic trigonometric cofunctions:

Cofunctions  

Sine and cosine \sin(\theta) = \cos(90^\circ-\theta)sin(θ)=cos(90  

∘

−θ)sine, left parenthesis, theta, right parenthesis, equals, cosine, left parenthesis, 90, degrees, minus, theta, right parenthesis

\cos(\theta) = \sin(90^\circ-\theta)cos(θ)=sin(90  

∘

−θ)cosine, left parenthesis, theta, right parenthesis, equals, sine, left parenthesis, 90, degrees, minus, theta, right parenthesis

Tangent and cotangent \tan(\theta) = \cot(90^\circ-\theta)tan(θ)=cot(90  

∘

−θ)tangent, left parenthesis, theta, right parenthesis, equals, cotangent, left parenthesis, 90, degrees, minus, theta, right parenthesis

\cot(\theta) = \tan(90^\circ-\theta)cot(θ)=tan(90  

∘

−θ)cotangent, left parenthesis, theta, right parenthesis, equals, tangent, left parenthesis, 90, degrees, minus, theta, right parenthesis

Secant and cosecant \sec(\theta) = \csc(90^\circ-\theta)sec(θ)=csc(90  

∘

−θ)\sec, left parenthesis, theta, right parenthesis, equals, \csc, left parenthesis, 90, degrees, minus, theta, right parenthesis

\csc(\theta) = \sec(90^\circ-\theta)csc(θ)=sec(90  

∘

−θ)\csc, left parenthesis, theta, right parenthesis, equals, \sec, left parenthesis, 90, degrees, minus, theta, right parenthesis

Neat! Whoever named the trig functions must have deeply understood the relationships between them.

Step-by-step explanation:

Answer 2

Answer:

1 is the answer for the given problem