State the other 5 trigonometric ratios for each acute angle T

Question

Cos T = 4/9

State the other 5 trigonometric ratios for each acute angle T

Answer 1

First, sketch the angle on the coordinate plane:

Next, based on the sketch and the Pythagorean Theorem, solve for "y":

y = √ (9² - 4²) = √(81 - 16) = √65

√65

sin(T) = opposite/hypotenuse = √65/9

tan(T) = opposite/adjacent = √65/4

csc(T) = hypotenuse/opposite = 9/√65 = 9√65/65

sec(T) = hypotenuse/adjacent = 9/4

cot(T) = adjacent/opposite = 4/√65 = 4√65/65

Answer 2

The 6 trig functions are:

sin = opp./hyp.

cos = adj./hyp.

tan = opp./adj.

csc = hyp./opp.

sec = hyp./adj.

cot = adj./opp.

Also, (hyp.)² = (opp.)² + (adj.)² and from the definition of cosine we know what adj. and hyp. are, so we can solve for the opp.:

cos T = 4/9

adj./hyp. = 4/9

adj. = 4 hyp. = 9

(hyp.)² = (opp.)² + (adj.)²

9² = (opp.)² + 4²

81 = (opp.)² + 16

81 - 16 = (opp.)²

65 = (opp.)²

opp. = √65

Now with these values for opp., adj. and hyp. we can determine all 6 trig functions.

Hope this helps!

Answer 3

Assuming that you are referring an Acute Angle in a Right Triangle

If Cos T = 4/9 and Cos = ADJ/HYP then

OPP = SQRT(9² - 4²) = SQRT(81 - 16) = SQRT(65)

OPP = SQRT(65) = 8.062

Cos T = 4/9

Sin T = 8.062/9

Tan T = 8.062/4

Sec T = 9/4

Csc T= 9/8.062

Cot T = 4/8.062

Now you try the other acute angle the angle opposite T in the Right Triangle.